Examkin

Simplifying Expressions and Combining Like Terms

34 soru

Tüm türler34Çoktan seçmeli18Sayısal7Eşleştirme9
Soru 1Soru →

Which of the following is equivalent to the expression 4x2(x−2y)−3x(x2−3xy)−(x3+xy2)4x^2(x - 2y) - 3x(x^2 - 3xy) - (x^3 + xy^2) for all real values of xx and yy?

Cevabı ve açıklamayı göster

Cevap: x2y−xy2x^2y - xy^2

Cevap

The correct simplified expression is x2y−xy2x^2y - xy^2.
Distributing the terms yields 4x3−8x2y−3x3+9x2y−x3−xy24x^3 - 8x^2y - 3x^3 + 9x^2y - x^3 - xy^2. Grouping and combining like terms results in (4−3−1)x3+(−8+9)x2y−xy2=x2y−xy2(4-3-1)x^3 + (-8+9)x^2y - xy^2 = x^2y - xy^2.

Adım Adım Çözüm

1
Distribute the term 4x24x^2 to the terms inside the first set of parentheses: 4x2(x−2y)4x^2(x - 2y).
4x3−8x2y4x^3 - 8x^2y
Multiplying a monomial by a binomial requires distributing the multiplier to each term inside.
2
Distribute the term −3x-3x to the terms inside the second set of parentheses: −3x(x2−3xy)-3x(x^2 - 3xy).
−3x3+9x2y-3x^3 + 9x^2y
Carefully apply the distributive property, remembering that multiplying a negative term by a negative term yields a positive term: −3x⋅(−3xy)=9x2y-3x \cdot (-3xy) = 9x^2y.
3
Distribute the negative sign (or −1-1) to the terms inside the third set of parentheses: −(x3+xy2)-(x^3 + xy^2).
−x3−xy2-x^3 - xy^2
Distributing the negative sign changes the signs of all terms inside the parentheses.
4
Combine all parts and group the like terms: x3x^3, x2yx^2y, and xy2xy^2.
x2y−xy2x^2y - xy^2
Add the coefficients of the terms with the same variable parts: (4−3−1)x3+(−8+9)x2y−xy2=0x3+x2y−xy2=x2y−xy2(4 - 3 - 1)x^3 + (-8 + 9)x^2y - xy^2 = 0x^3 + x^2y - xy^2 = x^2y - xy^2.

Anahtar Kavram

Simplifying algebraic expressions by distributing and combining like terms.
Tahmini Süre:1m 0s
Soru 2Soru →

For all real numbers xx and yy, which of the following is equivalent to the expression (x−2y)2−(x2−5xy)(x - 2y)^2 - (x^2 - 5xy)?

Cevabı ve açıklamayı göster

Cevap: 4y2+xy4y^2 + xy

Cevap

The expression is equivalent to 4y2+xy4y^2 + xy.
Expanding (x−2y)2(x - 2y)^2 yields x2−4xy+4y2x^2 - 4xy + 4y^2. Distributing the negative sign to −(x2−5xy)-(x^2 - 5xy) gives −x2+5xy-x^2 + 5xy. Combining the results yields (x2−x2)+(−4xy+5xy)+4y2=xy+4y2(x^2 - x^2) + (-4xy + 5xy) + 4y^2 = xy + 4y^2, which is equivalent to 4y2+xy4y^2 + xy.

Adım Adım Çözüm

1
Expand the squared binomial (x−2y)2(x - 2y)^2.
x2−4xy+4y2x^2 - 4xy + 4y^2
Apply the binomial squaring formula (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2 to expand the expression.
2
Distribute the negative sign across the second parenthetical expression −(x2−5xy)-(x^2 - 5xy).
−x2+5xy-x^2 + 5xy
Distributing the negative sign changes the signs of both terms inside the parentheses.
3
Combine like terms from the expanded parts.
xy+4y2xy + 4y^2
Group and add coefficients of the like terms: (x2−x2)+(−4xy+5xy)+4y2=0+xy+4y2(x^2 - x^2) + (-4xy + 5xy) + 4y^2 = 0 + xy + 4y^2.

Anahtar Kavram

Simplifying Expressions and Combining Like Terms
Soru 3Soru →

When the expression 6x−2x(x−3y)−(2x−12y)2−34x(6x−8)6x - 2x(x - 3y) - \left(2x - \frac{1}{2}y\right)^2 - \frac{3}{4}x(6x - 8) is simplified by combining like terms, what is the coefficient of the x2x^2 term?

Cevabı ve açıklamayı göster

Cevap: −212-\frac{21}{2}

Cevap

−212-\frac{21}{2}
Expanding the entire expression yields the x2x^2 terms −2x2-2x^2, −4x2-4x^2, and −92x2-\frac{9}{2}x^2. Summing these coefficients gives −2−4−92=−212-2 - 4 - \frac{9}{2} = -\frac{21}{2}. Therefore, the coefficient of the x2x^2 term is −212-\frac{21}{2}.

Adım Adım Çözüm

1
Expand the first parenthetical expression by distributing the term −2x-2x.
−2x(x−3y)=−2x2+6xy-2x(x - 3y) = -2x^2 + 6xy. The expression becomes: 6x−2x2+6xy−(2x−12y)2−34x(6x−8)6x - 2x^2 + 6xy - \left(2x - \frac{1}{2}y\right)^2 - \frac{3}{4}x(6x - 8).
Distribution is required to eliminate parentheses before terms can be combined.
2
Expand the squared binomial (2x−12y)2\left(2x - \frac{1}{2}y\right)^2 using the identity (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2, then distribute the negative sign.
(2x−12y)2=4x2−2xy+14y2\left(2x - \frac{1}{2}y\right)^2 = 4x^2 - 2xy + \frac{1}{4}y^2. Distributing the negative gives −4x2+2xy−14y2-4x^2 + 2xy - \frac{1}{4}y^2.
This simplifies the second parenthetical term of the expression.
3
Expand the third parenthetical term by distributing −34x-\frac{3}{4}x.
−34x(6x−8)=−92x2+6x-\frac{3}{4}x(6x - 8) = -\frac{9}{2}x^2 + 6x.
This simplifies the final parenthetical term of the expression.
4
Combine the coefficients of all the x2x^2 terms.
The x2x^2 terms are −2x2-2x^2, −4x2-4x^2, and −92x2-\frac{9}{2}x^2. Combining their coefficients gives: −2−4−92=−6−92=−122−92=−212-2 - 4 - \frac{9}{2} = -6 - \frac{9}{2} = -\frac{12}{2} - \frac{9}{2} = -\frac{21}{2}.
Combining like terms simplifies the expression to find the final coefficient of x2x^2.

Anahtar Kavram

Simplifying algebraic expressions by distributing coefficients and combining like terms.
Tahmini Süre:1m 30s
Soru 4Soru →

Simplify each of the algebraic expressions on the left by distributing and combining like terms, then match it with its equivalent simplified expression on the right.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Öğeler

−(x−2y)3−x(x+2y)(x−2y)+4y2(2x−y)-(x - 2y)^3 - x(x + 2y)(x - 2y) + 4y^2(2x - y)
2x(x−y)2−(x−2y)(x2+2xy+4y2)−2xy(x−2y)2x(x - y)^2 - (x - 2y)(x^2 + 2xy + 4y^2) - 2xy(x - 2y)
(x+y)3−(x−y)3−2y(3x2+y2)(x + y)^3 - (x - y)^3 - 2y(3x^2 + y^2)
(x2−y)2−(x2+y)(x2−y)−y(y−2x2)(x^2 - y)^2 - (x^2 + y)(x^2 - y) - y(y - 2x^2)

Eşleşmeler

Cevabı ve açıklamayı göster

Cevap

The expressions match as follows: the first expression matches −2x3+6x2y+4y3-2x^3 + 6x^2y + 4y^3; the second expression matches x3−6x2y+6xy2+8y3x^3 - 6x^2y + 6xy^2 + 8y^3; the third expression matches 00; and the fourth expression matches y2y^2.
Each expression on the left reduces to its matching counterpart on the right by carefully expanding terms (including cubes, squares, difference of squares, and difference of cubes) and combining like terms while correctly distributing negative signs.

Adım Adım Çözüm

1
Simplify the first expression by expanding each term individually.
E1=−x3+6x2y−12xy2+8y3−x3+4xy2+8xy2−4y3E_1 = -x^3 + 6x^2y - 12xy^2 + 8y^3 - x^3 + 4xy^2 + 8xy^2 - 4y^3
Expanding the binomial cube, the difference of squares product, and distributing the monomial allows us to identify like terms.
2
Combine like terms in the first expression.
E1=−2x3+6x2y+4y3E_1 = -2x^3 + 6x^2y + 4y^3
Combining the x3x^3, x2yx^2y, xy2xy^2, and y3y^3 terms simplifies the expression. The xy2xy^2 terms sum to zero.
3
Simplify the second expression by expanding each term individually.
E2=2x3−4x2y+2xy2−x3+8y3−2x2y+4xy2E_2 = 2x^3 - 4x^2y + 2xy^2 - x^3 + 8y^3 - 2x^2y + 4xy^2
Using algebraic expansion rules (including the difference of cubes product) exposes all individual terms.
4
Combine like terms in the second expression.
E2=x3−6x2y+6xy2+8y3E_2 = x^3 - 6x^2y + 6xy^2 + 8y^3
Adding coefficients of like variable terms gives the simplified form.
5
Simplify the third expression by expanding the cubic terms.
E3=(x3+3x2y+3xy2+y3)−(x3−3x2y+3xy2−y3)−(6x2y+2y3)=6x2y+2y3−6x2y−2y3E_3 = (x^3 + 3x^2y + 3xy^2 + y^3) - (x^3 - 3x^2y + 3xy^2 - y^3) - (6x^2y + 2y^3) = 6x^2y + 2y^3 - 6x^2y - 2y^3
Expanding the binomial cubes and distributing the negative signs shows that all terms cancel.
6
Combine like terms in the third expression.
E3=0E_3 = 0
All terms cancel out, leaving a final value of zero.
7
Simplify the fourth expression by expanding.
E4=x4−2x2y+y2−x4+y2−y2+2x2yE_4 = x^4 - 2x^2y + y^2 - x^4 + y^2 - y^2 + 2x^2y
Squaring the binomial, using the difference of squares, and distributing the negative variable simplifies the individual components.
8
Combine like terms in the fourth expression.
E4=y2E_4 = y^2
The x4x^4 and x2yx^2y terms cancel, and the y2y^2 terms simplify to y2y^2.

Anahtar Kavram

Simplifying Expressions and Combining Like Terms
Soru 5Soru →

When the expression −13x(x−2y)2−12x(xy−3y2)−(x3−2x2y)-\frac{1}{3}x(x - 2y)^2 - \frac{1}{2}x(xy - 3y^2) - (x^3 - 2x^2y) is fully simplified, what is the coefficient of the x2yx^2y term?

Cevabı ve açıklamayı göster

Cevap: 176\frac{17}{6}

Cevap

The coefficient of the x2yx^2y term is 176\frac{17}{6}.
By expanding all components of the expression: the first term yields −13x3+43x2y−43xy2-\frac{1}{3}x^3 + \frac{4}{3}x^2y - \frac{4}{3}xy^2, the second term yields −12x2y+32xy2-\frac{1}{2}x^2y + \frac{3}{2}xy^2, and the third term yields −x3+2x2y-x^3 + 2x^2y. Summing the coefficients of the x2yx^2y term gives 43−12+2=176\frac{4}{3} - \frac{1}{2} + 2 = \frac{17}{6}.

Adım Adım Çözüm

1
Expand the first term of the expression.
−13x(x−2y)2=−13x(x2−4xy+4y2)=−13x3+43x2y−43xy2-\frac{1}{3}x(x - 2y)^2 = -\frac{1}{3}x(x^2 - 4xy + 4y^2) = -\frac{1}{3}x^3 + \frac{4}{3}x^2y - \frac{4}{3}xy^2
Applying binomial expansion to (x−2y)2(x-2y)^2 and distributing −13x-\frac{1}{3}x.
2
Expand the second term of the expression.
−12x(xy−3y2)=−12x2y+32xy2-\frac{1}{2}x(xy - 3y^2) = -\frac{1}{2}x^2y + \frac{3}{2}xy^2
Distributing the term −12x-\frac{1}{2}x over the parenthetical terms.
3
Distribute the negative sign in the third term.
−(x3−2x2y)=−x3+2x2y-(x^3 - 2x^2y) = -x^3 + 2x^2y
Distributing the negative sign across all terms inside the parentheses.
4
Combine the coefficients of the x2yx^2y terms.
43−12+2=86−36+126=176\frac{4}{3} - \frac{1}{2} + 2 = \frac{8}{6} - \frac{3}{6} + \frac{12}{6} = \frac{17}{6}
Finding a common denominator of 6 to sum the coefficients of the x2yx^2y term.

Anahtar Kavram

Simplifying expressions by distributing terms, expanding binomials, and combining like terms with fractional coefficients.
Soru 6Soru →

For all real numbers xx and yy, match each algebraic expression on the left with its simplified equivalent expression on the right.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Öğeler

2(x−3y)+4y2(x - 3y) + 4y
−2(x−3y)+8y-2(x - 3y) + 8y
2(x+3y)−4y2(x + 3y) - 4y

Eşleşmeler

Cevabı ve açıklamayı göster

Cevap

The expression 2(x−3y)+4y2(x - 3y) + 4y simplifies to 2x−2y2x - 2y, the expression −2(x−3y)+8y-2(x - 3y) + 8y simplifies to −2x+14y-2x + 14y, and the expression 2(x+3y)−4y2(x + 3y) - 4y simplifies to 2x+2y2x + 2y.
Each expression on the left-hand side is expanded by applying the distributive property and then simplified by combining the terms involving yy. This correctly matches 2(x−3y)+4y2(x - 3y) + 4y to 2x−2y2x - 2y, −2(x−3y)+8y-2(x - 3y) + 8y to −2x+14y-2x + 14y, and 2(x+3y)−4y2(x + 3y) - 4y to 2x+2y2x + 2y.

Adım Adım Çözüm

1
Simplify the expression 2(x−3y)+4y2(x - 3y) + 4y.
2x−2y2x - 2y
Distribute 22 to both terms inside the parentheses to get 2x−6y2x - 6y, then combine the like terms −6y-6y and 4y4y to get −2y-2y.
2
Simplify the expression −2(x−3y)+8y-2(x - 3y) + 8y.
−2x+14y-2x + 14y
Distribute −2-2 to both terms inside the parentheses to get −2x+6y-2x + 6y, then combine the like terms 6y6y and 8y8y to get 14y14y.
3
Simplify the expression 2(x+3y)−4y2(x + 3y) - 4y.
2x+2y2x + 2y
Distribute 22 to both terms inside the parentheses to get 2x+6y2x + 6y, then combine the like terms 6y6y and −4y-4y to get 2y2y.

Anahtar Kavram

Simplifying algebraic expressions by distributing coefficients and combining like terms.
Tahmini Süre:1m 0s
Soru 7Soru →

For all real values of aa and bb, what is the simplified form of the expression 3a3b−2ab2−(a3b−5ab2)3a^3b - 2ab^2 - (a^3b - 5ab^2)?

Cevabı ve açıklamayı göster

Cevap: 2a3b+3ab22a^3b + 3ab^2

Cevap

2a3b+3ab22a^3b + 3ab^2
The correct expression is found by distributing the negative sign across the parentheses to change the signs of the terms inside, yielding −a3b+5ab2-a^3b + 5ab^2. Combining the like terms 3a3b3a^3b and −a3b-a^3b gives 2a3b2a^3b, and combining −2ab2-2ab^2 and +5ab2+5ab^2 gives 3ab23ab^2.

Adım Adım Çözüm

1
Distribute the negative sign to each term inside the parenthetical expression: −(a3b−5ab2)-(a^3b - 5ab^2).
−a3b+5ab2-a^3b + 5ab^2
To remove the parentheses, we multiply each term inside by −1-1.
2
Rewrite the full expression with the parentheses removed: 3a3b−2ab2−a3b+5ab23a^3b - 2ab^2 - a^3b + 5ab^2.
3a3b−2ab2−a3b+5ab23a^3b - 2ab^2 - a^3b + 5ab^2
This sets up the expression for combining like terms.
3
Group and combine the like terms: (3a3b−a3b)(3a^3b - a^3b) and (−2ab2+5ab2)(-2ab^2 + 5ab^2).
2a3b+3ab22a^3b + 3ab^2
Only terms with the exact same variable parts and exponents can be combined by adding or subtracting their coefficients.

Anahtar Kavram

Simplifying algebraic expressions by distributing negative signs and combining like terms.
Tahmini Süre:45s
Soru 8Soru →

Match each algebraic expression on the left with its equivalent simplified form on the right. Assume all variables represent real numbers.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Öğeler

2(3a2−4b)−3(a2−2b)2(3a^2 - 4b) - 3(a^2 - 2b)
a(3a−b)−2b(a−b)a(3a - b) - 2b(a - b)
(a+b)(3a−2b)−b2(a + b)(3a - 2b) - b^2
4a2−(a−b)(a+2b)−2b24a^2 - (a - b)(a + 2b) - 2b^2

Eşleşmeler

Cevabı ve açıklamayı göster

Cevap

The expression 2(3a2−4b)−3(a2−2b)2(3a^2 - 4b) - 3(a^2 - 2b) matches 3a2−2b3a^2 - 2b; a(3a−b)−2b(a−b)a(3a - b) - 2b(a - b) matches 3a2−3ab+2b23a^2 - 3ab + 2b^2; (a+b)(3a−2b)−b2(a + b)(3a - 2b) - b^2 matches 3a2+ab−3b23a^2 + ab - 3b^2; and 4a2−(a−b)(a+2b)−2b24a^2 - (a - b)(a + 2b) - 2b^2 matches 3a2−ab3a^2 - ab.
Each expression is correctly simplified by distributing coefficients, expanding binomial products, and collecting like terms.

Adım Adım Çözüm

1
Simplify the expression 2(3a2−4b)−3(a2−2b)2(3a^2 - 4b) - 3(a^2 - 2b)
3a2−2b3a^2 - 2b
Distribute the coefficients to remove the parentheses: 6a2−8b−3a2+6b6a^2 - 8b - 3a^2 + 6b. Group the a2a^2 terms and the bb terms, and then combine: (6−3)a2+(−8+6)b=3a2−2b(6 - 3)a^2 + (-8 + 6)b = 3a^2 - 2b.
2
Simplify the expression a(3a−b)−2b(a−b)a(3a - b) - 2b(a - b)
3a2−3ab+2b23a^2 - 3ab + 2b^2
Distribute the variables aa and −2b-2b: 3a2−ab−2ab+2b23a^2 - ab - 2ab + 2b^2. Combine the like terms −ab-ab and −2ab-2ab to get −3ab-3ab, resulting in 3a2−3ab+2b23a^2 - 3ab + 2b^2.
3
Simplify the expression (a+b)(3a−2b)−b2(a + b)(3a - 2b) - b^2
3a2+ab−3b23a^2 + ab - 3b^2
Multiply the binomial factors using the distributive property: (a+b)(3a−2b)=3a2−2ab+3ab−2b2=3a2+ab−2b2(a + b)(3a - 2b) = 3a^2 - 2ab + 3ab - 2b^2 = 3a^2 + ab - 2b^2. Subtract the remaining b2b^2 term: 3a2+ab−2b2−b2=3a2+ab−3b23a^2 + ab - 2b^2 - b^2 = 3a^2 + ab - 3b^2.
4
Simplify the expression 4a2−(a−b)(a+2b)−2b24a^2 - (a - b)(a + 2b) - 2b^2
3a2−ab3a^2 - ab
Expand (a−b)(a+2b)=a2+ab−2b2(a - b)(a + 2b) = a^2 + ab - 2b^2. Subtract this product from 4a24a^2 by distributing the negative sign: 4a2−a2−ab+2b24a^2 - a^2 - ab + 2b^2. Finally, subtract the last term 2b22b^2: 3a2−ab+2b2−2b2=3a2−ab3a^2 - ab + 2b^2 - 2b^2 = 3a^2 - ab.

Anahtar Kavram

Simplifying algebraic expressions by distributing factors and combining like terms
Tahmini Süre:2m 0s
Soru 9Soru →

For each algebraic expression on the left, match it to its completely simplified equivalent expression on the right by distributing terms and combining like terms.

Soldaki öğeye tıklayın, sonra eşleşen sağdaki öğeye tıklayın

Öğeler

2x(x2−3xy)−3y(x2−y2)−(2x3−6x2y)2x(x^2 - 3xy) - 3y(x^2 - y^2) - (2x^3 - 6x^2y)
(2x−y)3−8x(x2−3xy)−y3(2x - y)^3 - 8x(x^2 - 3xy) - y^3
x(2x−3y)2−y(x−2y)2−(4x3−13x2y)x(2x - 3y)^2 - y(x - 2y)^2 - (4x^3 - 13x^2y)
2x2(x−3y)−(x−y)3−y2(3x−y)2x^2(x - 3y) - (x - y)^3 - y^2(3x - y)

Eşleşmeler

Cevabı ve açıklamayı göster

Cevap

The expressions match as follows: the first simplifies to −3x2y+3y3-3x^2y + 3y^3; the second simplifies to 12x2y+6xy2−2y312x^2y + 6xy^2 - 2y^3; the third simplifies to 13xy2−4y313xy^2 - 4y^3; and the fourth simplifies to x3−3x2y−6xy2+2y3x^3 - 3x^2y - 6xy^2 + 2y^3.
Each expression is expanded fully by distributing multiplication and powers, then simplified by combining terms that have the exact same variable bases and exponents.

Adım Adım Çözüm

1
Simplify the first expression by distributing coefficients and combining like terms.
−3x2y+3y3-3x^2y + 3y^3
Expanding the expression gives 2x3−6x2y−3x2y+3y3−2x3+6x2y2x^3 - 6x^2y - 3x^2y + 3y^3 - 2x^3 + 6x^2y. Grouping the like terms: (2−2)x3+(−6−3+6)x2y+3y3(2 - 2)x^3 + (-6 - 3 + 6)x^2y + 3y^3, which simplifies to −3x2y+3y3-3x^2y + 3y^3.
2
Simplify the second expression using the binomial cube formula (a−b)3=a3−3a2b+3ab2−b3(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3.
12x2y+6xy2−2y312x^2y + 6xy^2 - 2y^3
Expanding (2x−y)3(2x - y)^3 yields 8x3−12x2y+6xy2−y38x^3 - 12x^2y + 6xy^2 - y^3. Subtracting the remaining terms gives 8x3−12x2y+6xy2−y3−8x3+24x2y−y38x^3 - 12x^2y + 6xy^2 - y^3 - 8x^3 + 24x^2y - y^3. Combining like terms yields 12x2y+6xy2−2y312x^2y + 6xy^2 - 2y^3.
3
Simplify the third expression by squaring the binomials and distributing.
13xy2−4y313xy^2 - 4y^3
First expand the squares: (2x−3y)2=4x2−12xy+9y2(2x - 3y)^2 = 4x^2 - 12xy + 9y^2 and (x−2y)2=x2−4xy+4y2(x - 2y)^2 = x^2 - 4xy + 4y^2. Distributing the variables yields 4x3−12x2y+9xy2−x2y+4xy2−4y3−4x3+13x2y4x^3 - 12x^2y + 9xy^2 - x^2y + 4xy^2 - 4y^3 - 4x^3 + 13x^2y. Combining like terms results in 13xy2−4y313xy^2 - 4y^3.
4
Simplify the fourth expression by expanding the cubic term and distributing signs.
x3−3x2y−6xy2+2y3x^3 - 3x^2y - 6xy^2 + 2y^3
Expand (x−y)3=x3−3x2y+3xy2−y3(x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3. Distribute all negative signs to get 2x3−6x2y−x3+3x2y−3xy2+y3−3xy2+y32x^3 - 6x^2y - x^3 + 3x^2y - 3xy^2 + y^3 - 3xy^2 + y^3. Combining like terms results in x3−3x2y−6xy2+2y3x^3 - 3x^2y - 6xy^2 + 2y^3.

Anahtar Kavram

Simplifying Expressions and Combining Like Terms
Soru 10Soru →

When the expression 3m(m−2n)2−(2m−n)(m2−3mn+2n2)−4n(m2−mn)3m(m - 2n)^2 - (2m - n)(m^2 - 3mn + 2n^2) - 4n(m^2 - mn) is fully simplified by combining like terms, what is the coefficient of the m2nm^2n term?

Cevabı ve açıklamayı göster

Cevap: −9-9

Cevap

−9-9
The correct answer is −9-9. This is found by carefully expanding each term and distributing the negative signs: the first term expands to 3m3−12m2n+12mn23m^3 - 12m^2n + 12mn^2; the second term, when subtracted, becomes −2m3+7m2n−7mn2+2n3-2m^3 + 7m^2n - 7mn^2 + 2n^3; and the third term simplifies to −4m2n+4mn2-4m^2n + 4mn^2. Summing the coefficients of the m2nm^2n terms gives −12+7−4=−9-12 + 7 - 4 = -9.

Adım Adım Çözüm

1
Expand the first term of the expression, 3m(m−2n)23m(m - 2n)^2.
First, square the binomial: (m−2n)2=m2−4mn+4n2(m - 2n)^2 = m^2 - 4mn + 4n^2. Next, distribute 3m3m to get 3m3−12m2n+12mn23m^3 - 12m^2n + 12mn^2.
To remove parentheses from the first term before combining like terms.
2
Expand the product in the second term, (2m−n)(m2−3mn+2n2)(2m - n)(m^2 - 3mn + 2n^2).
2m(m2−3mn+2n2)−n(m2−3mn+2n2)=2m3−6m2n+4mn2−m2n+3mn2−2n32m(m^2 - 3mn + 2n^2) - n(m^2 - 3mn + 2n^2) = 2m^3 - 6m^2n + 4mn^2 - m^2n + 3mn^2 - 2n^3. Combining like terms within this product yields 2m3−7m2n+7mn2−2n32m^3 - 7m^2n + 7mn^2 - 2n^3.
To expand the binomial-trinomial product before applying the subtraction.
3
Subtract the expanded second term and expand the third term, −4n(m2−mn)-4n(m^2 - mn).
Subtracting the second term gives −2m3+7m2n−7mn2+2n3-2m^3 + 7m^2n - 7mn^2 + 2n^3. Distributing the negative sign in the third term gives −4m2n+4mn2-4m^2n + 4mn^2.
To distribute the negative signs across the remaining parenthetical expressions.
4
Identify and combine all the m2nm^2n terms to find the final coefficient.
The m2nm^2n terms are −12m2n-12m^2n (from the first term), +7m2n+7m^2n (from the subtracted second term), and −4m2n-4m^2n (from the third term). Combining these yields (−12+7−4)m2n=−9m2n(-12 + 7 - 4)m^2n = -9m^2n.
To determine the final coefficient of the m2nm^2n term.

Anahtar Kavram

Simplifying Expressions and Combining Like Terms
Tahmini Süre:2m 0s
Soru 11Soru →

When the expression 5(2x−3y)−3(x−4y)5(2x - 3y) - 3(x - 4y) is simplified, what is the coefficient of yy?

Cevabı ve açıklamayı göster

Cevap: -3

Cevap

The coefficient of yy is −3-3.
Distributing 55 to (2x−3y)(2x - 3y) yields 10x−15y10x - 15y. Distributing −3-3 to (x−4y)(x - 4y) yields −3x+12y-3x + 12y. Combining the yy terms gives −15y+12y=−3y-15y + 12y = -3y. Therefore, the coefficient of yy is −3-3.

Adım Adım Çözüm

1
Distribute the coefficients outside the parentheses.
10x−15y−3x+12y10x - 15y - 3x + 12y
To eliminate the parentheses so that like terms can be combined.
2
Group and combine the terms containing yy.
−15y+12y=−3y-15y + 12y = -3y
To determine the final simplified term containing yy and identify its coefficient.

Anahtar Kavram

Distributing terms (especially negative coefficients) and combining like terms.
Tahmini Süre:45s
Soru 12Soru →

For all real numbers xx and yy, when the expression −2x(x−3y)2−y2(4x−y)−3x2(2x−y)-2x(x - 3y)^2 - y^2(4x - y) - 3x^2(2x - y) is completely simplified, what is the coefficient of the x2yx^2y term?

Cevabı ve açıklamayı göster

Cevap: 15

Cevap

15
Completely simplifying the given expression yields −8x3+15x2y−22xy2+y3-8x^3 + 15x^2y - 22xy^2 + y^3. The coefficient of the x2yx^2y term is 1515, which is obtained by combining the term 12x2y12x^2y (from the distribution of −2x-2x over the middle term of the squared binomial) and the term 3x2y3x^2y (from the distribution of −3x2-3x^2 over −y-y).

Adım Adım Çözüm

1
Expand the binomial squared expression (x−3y)2(x - 3y)^2.
(x−3y)2=x2−6xy+9y2(x - 3y)^2 = x^2 - 6xy + 9y^2
Following the order of operations, we must square the binomial before distributing the outer term.
2
Distribute −2x-2x to the trinomial result from the previous step.
−2x(x2−6xy+9y2)=−2x3+12x2y−18xy2-2x(x^2 - 6xy + 9y^2) = -2x^3 + 12x^2y - 18xy^2
Using the distributive property, we multiply coefficients and add exponents of like bases (noting that −2x×−6xy=12x2y-2x \times -6xy = 12x^2y).
3
Distribute −y2-y^2 to the binomial (4x−y)(4x - y).
−y2(4x−y)=−4xy2+y3-y^2(4x - y) = -4xy^2 + y^3
We multiply each term inside the parentheses by −y2-y^2, keeping track of the signs.
4
Distribute −3x2-3x^2 to the binomial (2x−y)(2x - y).
−3x2(2x−y)=−6x3+3x2y-3x^2(2x - y) = -6x^3 + 3x^2y
We distribute the −3x2-3x^2 factor, ensuring that multiplying two negative values results in a positive term (−3x2×−y=3x2y-3x^2 \times -y = 3x^2y).
5
Combine the coefficients of all like terms containing x2yx^2y.
12x2y+3x2y=15x2y12x^2y + 3x^2y = 15x^2y
We add the coefficients of the terms that share the exact variable part x2yx^2y to find the final coefficient.

Anahtar Kavram

Simplifying algebraic expressions by distributing terms, applying exponent rules, and combining like terms.
Soru 13Soru →

For all real values of aa and bb, which of the following is equivalent to the expression 2a(a2−3b)−3(a3−2ab+b2)−(4b2−a3)2a(a^2 - 3b) - 3(a^3 - 2ab + b^2) - (4b^2 - a^3)?

Cevabı ve açıklamayı göster

Cevap: −7b2-7b^2

Cevap

−7b2-7b^2
Distributing the terms yields 2a3−6ab−3a3+6ab−3b2−4b2+a32a^3 - 6ab - 3a^3 + 6ab - 3b^2 - 4b^2 + a^3. Combining the like terms for a3a^3 gives 2a3−3a3+a3=02a^3 - 3a^3 + a^3 = 0. Combining the abab terms gives −6ab+6ab=0-6ab + 6ab = 0. Combining the b2b^2 terms gives −3b2−4b2=−7b2-3b^2 - 4b^2 = -7b^2. Thus, the simplified expression is −7b2-7b^2.

Adım Adım Çözüm

1
Distribute the factors outside the parentheses to each term inside the parentheses.
The terms expand to: 2a(a2)−2a(3b)−3(a3)−3(−2ab)−3(b2)−1(4b2)−1(−a3)=2a3−6ab−3a3+6ab−3b2−4b2+a32a(a^2) - 2a(3b) - 3(a^3) - 3(-2ab) - 3(b^2) - 1(4b^2) - 1(-a^3) = 2a^3 - 6ab - 3a^3 + 6ab - 3b^2 - 4b^2 + a^3
Distribution eliminates parentheses, making it possible to group and combine like terms.
2
Group like terms together based on their variable parts and powers.
(2a3−3a3+a3)+(−6ab+6ab)+(−3b2−4b2)(2a^3 - 3a^3 + a^3) + (-6ab + 6ab) + (-3b^2 - 4b^2)
Grouping like terms makes it easier to perform the arithmetic on the coefficients.
3
Combine the coefficients for each group of like terms.
0a3+0ab−7b2=−7b20a^3 + 0ab - 7b^2 = -7b^2
Simplifying the coefficient sums yields the final simplified form of the expression.

Anahtar Kavram

Simplifying expressions by distributing terms and combining like terms
Soru 14Soru →

For all real values of xx and yy, the expression 3x(x−2y)−2(x2−4xy+y2)−x23x(x - 2y) - 2(x^2 - 4xy + y^2) - x^2 can be written in the form axy+by2axy + by^2, where aa and bb are constants. What is the value of aa?

Cevabı ve açıklamayı göster

Cevap: 2

Cevap

The value of aa is 2.
The value of aa is 2 because distributing 3x(x−2y)3x(x - 2y) yields 3x2−6xy3x^2 - 6xy, and distributing −2(x2−4xy+y2)-2(x^2 - 4xy + y^2) yields −2x2+8xy−2y2-2x^2 + 8xy - 2y^2. Combining these with the −x2-x^2 term yields (3−2−1)x2+(−6+8)xy−2y2=2xy−2y2(3-2-1)x^2 + (-6+8)xy - 2y^2 = 2xy - 2y^2. Comparing this to axy+by2axy + by^2 shows that aa, the coefficient of the xyxy term, is 2.

Adım Adım Çözüm

1
Distribute 3x3x across the first parenthetical expression (x−2y)(x - 2y)
3x2−6xy3x^2 - 6xy
To clear the first set of parentheses by multiplying 3x3x by each term inside.
2
Distribute −2-2 across the second parenthetical expression (x2−4xy+y2)(x^2 - 4xy + y^2)
−2x2+8xy−2y2-2x^2 + 8xy - 2y^2
To clear the second set of parentheses. Note that multiplying −2-2 by −4xy-4xy yields a positive term +8xy+8xy due to the sign rules.
3
Write the full expression and group like terms
(3x2−2x2−x2)+(−6xy+8xy)−2y2(3x^2 - 2x^2 - x^2) + (-6xy + 8xy) - 2y^2
To group terms with identical variable parts so they can be combined.
4
Combine the coefficients of the grouped terms
2xy−2y22xy - 2y^2
Simplifying the groups: 3−2−1=03-2-1=0 for the x2x^2 terms, and −6+8=2-6+8=2 for the xyxy terms.
5
Compare the simplified expression to the form axy+by2axy + by^2 to find the coefficient aa
a=2a = 2
The coefficient of the xyxy term is 22, which corresponds to aa in the target expression.

Anahtar Kavram

Simplifying Expressions and Combining Like Terms
Tahmini Süre:1m 30s
Soru 15Soru →

For all real values of xx and yy, which of the following is equivalent to the expression x(x−y)2−x2(x−2y)x(x - y)^2 - x^2(x - 2y)?

Cevabı ve açıklamayı göster

Cevap: xy2xy^2

Cevap

The simplified expression is xy2xy^2.
Expanding (x−y)2(x - y)^2 yields x2−2xy+y2x^2 - 2xy + y^2. Distributing xx to this expression results in x3−2x2y+xy2x^3 - 2x^2y + xy^2. Distributing −x2-x^2 to (x−2y)(x - 2y) yields −x3+2x2y-x^3 + 2x^2y. Combining these parts gives (x3−x3)+(−2x2y+2x2y)+xy2(x^3 - x^3) + (-2x^2y + 2x^2y) + xy^2, which simplifies completely to xy2xy^2.

Adım Adım Çözüm

1
Expand the squared binomial (x−y)2(x - y)^2 using the algebraic identity (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2.
(x−y)2=x2−2xy+y2(x - y)^2 = x^2 - 2xy + y^2
Expanding the binomial is necessary before distributing the outer variable.
2
Distribute the term xx to each term in the expanded binomial, and distribute the term −x2-x^2 to each term inside the second parenthesis.
x(x2−2xy+y2)=x3−2x2y+xy2x(x^2 - 2xy + y^2) = x^3 - 2x^2y + xy^2 and −x2(x−2y)=−x3+2x2y-x^2(x - 2y) = -x^3 + 2x^2y
Distribution eliminates parentheses and prepares the expression for combining like terms.
3
Combine all like terms in the resulting expression: (x3−x3)+(−2x2y+2x2y)+xy2(x^3 - x^3) + (-2x^2y + 2x^2y) + xy^2.
xy2xy^2
Combining like terms simplifies the expression to its final equivalent form.

Anahtar Kavram

Simplifying algebraic expressions by expanding binomials, distributing variables, and combining like terms.
Tahmini Süre:1m 0s
Soru 16Soru →

When the expression −(p−2q)3−p(p−3q)(2p+q)+3q2(p−q)-(p - 2q)^3 - p(p - 3q)(2p + q) + 3q^2(p - q) is completely simplified by combining like terms, what is the coefficient of p2qp^2q?

Cevabı ve açıklamayı göster

Cevap: 11

Cevap

The coefficient of p2qp^2q in the fully simplified expression is 11.
Expanding the three components of the expression yields: −(p−2q)3=−p3+6p2q−12pq2+8q3-(p - 2q)^3 = -p^3 + 6p^2q - 12pq^2 + 8q^3; −p(p−3q)(2p+q)=−2p3+5p2q+3pq2-p(p - 3q)(2p + q) = -2p^3 + 5p^2q + 3pq^2; and 3q2(p−q)=3pq2−3q33q^2(p - q) = 3pq^2 - 3q^3. Combining these terms gives the simplified polynomial −3p3+11p2q−6pq2+5q3-3p^3 + 11p^2q - 6pq^2 + 5q^3. The coefficient of p2qp^2q is 11.

Adım Adım Çözüm

1
Expand and negate the term −(p−2q)3-(p - 2q)^3
−p3+6p2q−12pq2+8q3-p^3 + 6p^2q - 12pq^2 + 8q^3
Using the binomial theorem expansion for (a−b)3=a3−3a2b+3ab2−b3(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 and distributing the negative sign.
2
Multiply and distribute −p(p−3q)(2p+q)-p(p - 3q)(2p + q)
−2p3+5p2q+3pq2-2p^3 + 5p^2q + 3pq^2
First multiply the binomials (p−3q)(2p+q)=2p2−5pq−3q2(p - 3q)(2p + q) = 2p^2 - 5pq - 3q^2, and then multiply each term by −p-p.
3
Distribute 3q2(p−q)3q^2(p - q)
3pq2−3q33pq^2 - 3q^3
Multiply 3q23q^2 by both terms inside the binomial.
4
Combine the like terms of p2qp^2q
1111
Identify and sum all terms containing p2qp^2q: 6p2q+5p2q=11p2q6p^2q + 5p^2q = 11p^2q.

Anahtar Kavram

Simplifying algebraic expressions by expanding polynomials, applying the distributive property with negative signs, and combining like terms.
Soru 17Soru →

When the expression 2x(x2−3xy+2y2)−(x−2y)3−y(x2−5xy+4y2)2x(x^2 - 3xy + 2y^2) - (x - 2y)^3 - y(x^2 - 5xy + 4y^2) is fully simplified by combining like terms, what is the coefficient of the xy2xy^2 term?

Cevabı ve açıklamayı göster

Cevap: -3

Cevap

The coefficient of the xy2xy^2 term is −3-3.
Expanding the terms of the expression yields 2x3−6x2y+4xy22x^3 - 6x^2y + 4xy^2, −x3+6x2y−12xy2+8y3-x^3 + 6x^2y - 12xy^2 + 8y^3, and −x2y+5xy2−4y3-x^2y + 5xy^2 - 4y^3. Summing the coefficients of the xy2xy^2 terms gives 4−12+5=−34 - 12 + 5 = -3. Thus, the coefficient of xy2xy^2 is −3-3.

Adım Adım Çözüm

1
Expand the first term of the expression: 2x(x2−3xy+2y2)2x(x^2 - 3xy + 2y^2)
2x3−6x2y+4xy22x^3 - 6x^2y + 4xy^2
Distribute the monomial 2x2x to each term inside the parentheses: 2x⋅x2=2x32x \cdot x^2 = 2x^3, 2x⋅(−3xy)=−6x2y2x \cdot (-3xy) = -6x^2y, and 2x⋅2y2=4xy22x \cdot 2y^2 = 4xy^2.
2
Expand the cubed binomial (x−2y)3(x - 2y)^3 and distribute the negative sign
−x3+6x2y−12xy2+8y3-x^3 + 6x^2y - 12xy^2 + 8y^3
First, expand the binomial (x−2y)3=x3−3(x2)(2y)+3(x)(2y)2−(2y)3=x3−6x2y+12xy2−8y3(x - 2y)^3 = x^3 - 3(x^2)(2y) + 3(x)(2y)^2 - (2y)^3 = x^3 - 6x^2y + 12xy^2 - 8y^3. Then, distribute the negative sign to all terms inside the parentheses.
3
Expand the third term: −y(x2−5xy+4y2)-y(x^2 - 5xy + 4y^2)
−x2y+5xy2−4y3-x^2y + 5xy^2 - 4y^3
Distribute the negative monomial −y-y to each term inside the parentheses: −y⋅x2=−x2y-y \cdot x^2 = -x^2y, −y⋅(−5xy)=5xy2-y \cdot (-5xy) = 5xy^2, and −y⋅4y2=−4y3-y \cdot 4y^2 = -4y^3.
4
Group and combine all like terms
x3−x2y−3xy2+4y3x^3 - x^2y - 3xy^2 + 4y^3
Combine the coefficients of the corresponding terms: (2x3−x3)+(−6x2y+6x2y−x2y)+(4xy2−12xy2+5xy2)+(8y3−4y3)=x3−x2y−3xy2+4y3(2x^3 - x^3) + (-6x^2y + 6x^2y - x^2y) + (4xy^2 - 12xy^2 + 5xy^2) + (8y^3 - 4y^3) = x^3 - x^2y - 3xy^2 + 4y^3.

Anahtar Kavram

Simplifying Expressions and Combining Like Terms
Tahmini Süre:3m 0s
Soru 18Soru →

When the expression 2x2(3x−y)−3y(x2−2y)−(x3−3xy2)2x^2(3x - y) - 3y(x^2 - 2y) - (x^3 - 3xy^2) is fully simplified by combining like terms, which of the following represents the resulting expression?

Cevabı ve açıklamayı göster

Cevap: 5x3−5x2y+3xy2+6y25x^3 - 5x^2y + 3xy^2 + 6y^2

Cevap

5x3−5x2y+3xy2+6y25x^3 - 5x^2y + 3xy^2 + 6y^2
The correct expression is obtained by systematically distributing the coefficients outside the parentheses and then grouping and combining the coefficients of the like terms: (6x3−x3)+(−2x2y−3x2y)+3xy2+6y2=5x3−5x2y+3xy2+6y2(6x^3 - x^3) + (-2x^2y - 3x^2y) + 3xy^2 + 6y^2 = 5x^3 - 5x^2y + 3xy^2 + 6y^2.

Adım Adım Çözüm

1
Distribute 2x22x^2 to each term in the first parenthetical expression: (3x−y)(3x - y)
6x3−2x2y6x^3 - 2x^2y
Applying the distributive property of multiplication over subtraction.
2
Distribute −3y-3y to each term in the second parenthetical expression: (x2−2y)(x^2 - 2y)
−3x2y+6y2-3x^2y + 6y^2
Applying the distributive property and multiplying negative coefficients (−3×−2=6-3 \times -2 = 6).
3
Distribute the negative sign to each term in the third parenthetical expression: (x3−3xy2)(x^3 - 3xy^2)
−x3+3xy2-x^3 + 3xy^2
Distributing −1-1 across the parentheses to remove them.
4
Combine the expanded expressions and group the like terms
(6x3−x3)+(−2x2y−3x2y)+3xy2+6y2(6x^3 - x^3) + (-2x^2y - 3x^2y) + 3xy^2 + 6y^2
Grouping together terms that have the same variables raised to the same powers.
5
Simplify by performing the operations on the coefficients of the like terms
5x3−5x2y+3xy2+6y25x^3 - 5x^2y + 3xy^2 + 6y^2
Combining coefficients: 6−1=56 - 1 = 5 for the x3x^3 terms, and −2−3=−5-2 - 3 = -5 for the x2yx^2y terms.

Anahtar Kavram

Simplifying algebraic expressions by distributing coefficients and combining like terms.

Alternatif Yöntem

To check your work, substitute simple values for xx and yy, such as x=1x = 1 and y=1y = 1, into the original expression and the simplified expression. Evaluating the original expression: 2(1)2(3(1)−1)−3(1)(12−2(1))−(13−3(1)(1)2)=2(2)−3(−1)−(1−3)=4+3−(−2)=92(1)^2(3(1) - 1) - 3(1)(1^2 - 2(1)) - (1^3 - 3(1)(1)^2) = 2(2) - 3(-1) - (1 - 3) = 4 + 3 - (-2) = 9. Evaluating the correct simplified expression: 5(1)3−5(1)2(1)+3(1)(1)2+6(1)2=5−5+3+6=95(1)^3 - 5(1)^2(1) + 3(1)(1)^2 + 6(1)^2 = 5 - 5 + 3 + 6 = 9. Since both evaluations yield 9, this confirms the simplification.
Tahmini Süre:1m 0s
Soru 19Soru →

A business analyst models the daily cost, CC, and daily revenue, RR, of a manufacturing process using the following expressions:

C=2x2(x−3y)−y(x2−2y2)C = 2x^2(x - 3y) - y(x^2 - 2y^2)
R=5x3−x(y−2x)2+3y3R = 5x^3 - x(y - 2x)^2 + 3y^3

where xx represents the number of units of product X sold, and yy represents the number of units of product Y sold. The daily profit, PP, is defined as P=R−CP = R - C. When the profit expression is fully simplified by combining like terms, which of the following expressions represents the daily profit PP?

Cevabı ve açıklamayı göster

Cevap: $-x^3 + 11x^2y - xy^2 + y^3

Cevap

−x3+11x2y−xy2+y3-x^3 + 11x^2y - xy^2 + y^3
The expression −x3+11x2y−xy2+y3-x^3 + 11x^2y - xy^2 + y^3 is correct. Calculating profit requires subtracting the fully simplified cost from the fully simplified revenue. Simplifying CC yields 2x3−7x2y+2y32x^3 - 7x^2y + 2y^3. Expanding and simplifying RR yields x3+4x2y−xy2+3y3x^3 + 4x^2y - xy^2 + 3y^3. Subtracting these two expressions and correctly distributing the negative sign gives −x3+11x2y−xy2+y3-x^3 + 11x^2y - xy^2 + y^3.

Adım Adım Çözüm

1
Expand and simplify the cost expression C=2x2(x−3y)−y(x2−2y2)C = 2x^2(x - 3y) - y(x^2 - 2y^2).
C=2x3−6x2y−x2y+2y3=2x3−7x2y+2y3C = 2x^3 - 6x^2y - x^2y + 2y^3 = 2x^3 - 7x^2y + 2y^3
Distribute the terms outside the parentheses and combine the like terms −6x2y-6x^2y and −x2y-x^2y.
2
Expand and simplify the revenue expression R=5x3−x(y−2x)2+3y3R = 5x^3 - x(y - 2x)^2 + 3y^3.
R=5x3−x(y2−4xy+4x2)+3y3=5x3−xy2+4x2y−4x3+3y3=x3+4x2y−xy2+3y3R = 5x^3 - x(y^2 - 4xy + 4x^2) + 3y^3 = 5x^3 - xy^2 + 4x^2y - 4x^3 + 3y^3 = x^3 + 4x^2y - xy^2 + 3y^3
Square the binomial (y−2x)2=y2−4xy+4x2(y - 2x)^2 = y^2 - 4xy + 4x^2, distribute the −x-x, and combine the like terms 5x35x^3 and −4x3-4x^3.
3
Subtract the cost expression from the revenue expression to find P=R−CP = R - C.
P=(x3+4x2y−xy2+3y3)−(2x3−7x2y+2y3)=x3+4x2y−xy2+3y3−2x3+7x2y−2y3=−x3+11x2y−xy2+y3P = (x^3 + 4x^2y - xy^2 + 3y^3) - (2x^3 - 7x^2y + 2y^3) = x^3 + 4x^2y - xy^2 + 3y^3 - 2x^3 + 7x^2y - 2y^3 = -x^3 + 11x^2y - xy^2 + y^3
Distribute the subtraction negative sign to all terms in the cost expression and group like terms together to obtain the final simplified expression.

Anahtar Kavram

Simplifying multi-variable algebraic expressions by distributing coefficients and combining like terms.
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Match each algebraic expression on the left with its fully simplified equivalent expression on the right. All variables represent real numbers.

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Öğeler

3a(a2−2ab)−(a3−5a2b)3a(a^2 - 2ab) - (a^3 - 5a^2b)
a2(2a−b)−2a(a2−ab)a^2(2a - b) - 2a(a^2 - ab)
−(a3+3a2b)+3a2(a+b)-(a^3 + 3a^2b) + 3a^2(a + b)

Eşleşmeler

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Cevap

The first expression 3a(a2−2ab)−(a3−5a2b)3a(a^2 - 2ab) - (a^3 - 5a^2b) matches 2a3−a2b2a^3 - a^2b. The second expression a2(2a−b)−2a(a2−ab)a^2(2a - b) - 2a(a^2 - ab) matches a2ba^2b. The third expression −(a3+3a2b)+3a2(a+b)-(a^3 + 3a^2b) + 3a^2(a + b) matches 2a32a^3.
Each expression on the left is simplified by distributing the coefficients and combining the like terms. The first expression simplifies to 2a3−a2b2a^3 - a^2b. The second expression simplifies to a2ba^2b. The third expression simplifies to 2a32a^3. These match the corresponding simplified expressions on the right.

Adım Adım Çözüm

1
Simplify the first expression 3a(a2−2ab)−(a3−5a2b)3a(a^2 - 2ab) - (a^3 - 5a^2b)
2a3−a2b2a^3 - a^2b
Multiply 3a3a by both terms in the first parentheses to get 3a3−6a2b3a^3 - 6a^2b. Distribute the negative sign to both terms in the second parentheses to get −a3+5a2b-a^3 + 5a^2b. Combine the a3a^3 terms to get 2a32a^3 and the a2ba^2b terms to get −a2b-a^2b.
2
Simplify the second expression a2(2a−b)−2a(a2−ab)a^2(2a - b) - 2a(a^2 - ab)
a2ba^2b
Multiply a2a^2 by both terms in the first parentheses to get 2a3−a2b2a^3 - a^2b. Multiply −2a-2a by both terms in the second parentheses to get −2a3+2a2b-2a^3 + 2a^2b. Combine the a3a^3 terms to get 00 and the a2ba^2b terms to get a2ba^2b.
3
Simplify the third expression −(a3+3a2b)+3a2(a+b)-(a^3 + 3a^2b) + 3a^2(a + b)
2a32a^3
Distribute the negative sign to get −a3−3a2b-a^3 - 3a^2b. Multiply 3a23a^2 by both terms in the second parentheses to get 3a3+3a2b3a^3 + 3a^2b. Combine the a3a^3 terms to get 2a32a^3 and the a2ba^2b terms to get 00.

Anahtar Kavram

Simplifying Expressions and Combining Like Terms
Tahmini Süre:1m 30s
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