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Simplifying Algebraic Expressions

33 questions

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Question 1Question →

Simplify the following algebraic expression completely. Write your final answer in standard form (highest to lowest power):

−2(n2−4n+3)+3n(n−2)-2(n^2 - 4n + 3) + 3n(n - 2)
Show answer & explanation

Answer: n^2 + 2n - 6; n^2+2n-6; 1n^2 + 2n - 6; n^2 + 2n -6; n^2+2n - 6; 1n^2+2n-6

Answer

n^2 + 2n - 6
The completely simplified form is determined by first distributing the multipliers into their respective parentheses. For the first group, -2(n^2 - 4n + 3) expands to -2n^2 + 8n - 6. For the second group, 3n(n - 2) expands to 3n^2 - 6n. Combining the expansions results in -2n^2 + 8n - 6 + 3n^2 - 6n. By adding the coefficients of like terms, the final standard form evaluates to n^2 + 2n - 6.

Step-by-Step Solution

1
Distribute -2 into the first polynomial, (n^2 - 4n + 3).
-2n^2 + 8n - 6
The distributive property requires multiplying the outside term by every term inside the parentheses. Care must be taken with the negative sign: -2 multiplied by -4n becomes positive 8n.
2
Distribute 3n into the second binomial, (n - 2).
3n^2 - 6n
Using the distributive property again, multiplying 3n by n gives 3n^2, and 3n multiplied by -2 gives -6n.
3
Group the like terms from both expansions together.
(-2n^2 + 3n^2) + (8n - 6n) - 6
Like terms are terms that share the exact same variables raised to the exact same power. Grouping them helps systematically combine the coefficients.
4
Simplify the grouped terms.
n^2 + 2n - 6
Adding the coefficients of the grouped terms: -2 + 3 = 1 for the n^2 terms, and 8 - 6 = 2 for the n terms.

Key Concept

Simplifying algebraic expressions via polynomial expansion and combining like terms.
Question 2Question →

Which simplified algebraic expression corresponds to each unsimplified expression? Match each item on the left with its correctly simplified equivalent on the right.

Click a left item, then click its matching right item

Items

4x+3x−54x + 3x - 5
2(3x+4)2(3x + 4)
9x−4x+29x - 4x + 2
3(x−2)+53(x - 2) + 5

Matches

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Answer

The correct pairings are: 4x+3x−54x + 3x - 5 matches with 7x−57x - 5; 2(3x+4)2(3x + 4) matches with 6x+86x + 8; 9x−4x+29x - 4x + 2 matches with 5x+25x + 2; and 3(x−2)+53(x - 2) + 5 matches with 3x−13x - 1.
Each expression is correctly matched to its simplified equivalent by strictly applying the distributive property to eliminate any grouping symbols, followed by combining like terms (terms with identical variables) and grouping constants.

Step-by-Step Solution

1
Simplify the first expression, 4x+3x−54x + 3x - 5.
7x−57x - 5
Combining the like variable terms (4x4x and 3x3x) gives 7x7x. The constant −5-5 remains unchanged.
2
Simplify the second expression, 2(3x+4)2(3x + 4).
6x+86x + 8
Applying the distributive property multiplies each term inside the parentheses by 22.
3
Simplify the third expression, 9x−4x+29x - 4x + 2.
5x+25x + 2
Combining the like variable terms (9x9x and −4x-4x) gives 5x5x. The constant 22 remains unchanged.
4
Simplify the fourth expression, 3(x−2)+53(x - 2) + 5.
3x−13x - 1
Distributing the 33 yields 3x−63x - 6. Combining the constant terms (−6-6 and 55) results in −1-1.

Key Concept

Simplifying Algebraic Expressions by Combining Like Terms and Distributing
Question 3Question →

Match each unsimplified algebraic expression with its correctly simplified polynomial equivalent.

Click a left item, then click its matching right item

Items

3x−2(x−5)+43x - 2(x - 5) + 4
5(2x2−x)−3x(x+2)5(2x^2 - x) - 3x(x + 2)
4x−[2x+3(x−1)]4x - [2x + 3(x - 1)]
12(4x−6)+3(x+2)\frac{1}{2}(4x - 6) + 3(x + 2)

Matches

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Answer

The correct matches pair each unsimplified expression with its corresponding standard form by applying the distributive property and carefully combining like terms.
Each correct pairing is derived by rigorously applying the distributive property, adhering to the order of operations for nested groupings, and grouping matching polynomial degrees correctly.

Step-by-Step Solution

1
Simplify the first expression: 3x−2(x−5)+43x - 2(x - 5) + 4.
x+14x + 14
Apply the distributive property to −2(x−5)-2(x - 5) to get −2x+10-2x + 10. Then combine like terms: (3x−2x)+(10+4)(3x - 2x) + (10 + 4).
2
Simplify the second expression: 5(2x2−x)−3x(x+2)5(2x^2 - x) - 3x(x + 2).
7x2−11x7x^2 - 11x
Distribute 55 to get 10x2−5x10x^2 - 5x, and distribute −3x-3x to get −3x2−6x-3x^2 - 6x. Group like terms: (10x2−3x2)+(−5x−6x)(10x^2 - 3x^2) + (-5x - 6x).
3
Simplify the third expression: 4x−[2x+3(x−1)]4x - [2x + 3(x - 1)].
−x+3-x + 3
Follow the order of operations by simplifying the innermost grouping first: 3(x−1)=3x−33(x - 1) = 3x - 3. The brackets become [5x−3][5x - 3]. Distributing the subtraction gives 4x−5x+34x - 5x + 3.
4
Simplify the fourth expression: 12(4x−6)+3(x+2)\frac{1}{2}(4x - 6) + 3(x + 2).
5x+35x + 3
Multiply coefficients: 12(4x)−12(6)=2x−3\frac{1}{2}(4x) - \frac{1}{2}(6) = 2x - 3, and 3(x)+3(2)=3x+63(x) + 3(2) = 3x + 6. Combine xx terms and constants: (2x+3x)+(−3+6)(2x + 3x) + (-3 + 6).

Key Concept

Simplifying Algebraic Expressions
Question 4Question →

Which of the following is the simplified form of the expression 4(2x−3)+5x4(2x - 3) + 5x?

Show answer & explanation

Answer: 13x−1213x - 12

Answer

13x−1213x - 12
The correct answer is obtained by first applying the distributive property to multiply 44 by both 2x2x and −3-3, yielding 8x−128x - 12. Then, the like terms 8x8x and 5x5x are combined to get 13x13x. The constant −12-12 cannot be combined with the variable terms, so the fully simplified expression is 13x−1213x - 12.

Step-by-Step Solution

1
Distribute the 4 into the parentheses.
8x−12+5x8x - 12 + 5x
The distributive property requires multiplying the constant outside the parentheses by each term inside.
2
Group and combine the like terms.
13x−1213x - 12
The terms 8x8x and 5x5x are like terms and can be added together. The constant −12-12 cannot be combined with variable terms.

Key Concept

Simplifying Algebraic Expressions via Distribution and Combining Like Terms
Question 5Question →

Which of the following is equivalent to the expression 8−2(3x−4)+5x8 - 2(3x - 4) + 5x?

Show answer & explanation

Answer: −x+16-x + 16

Answer

The correct simplified expression is −x+16-x + 16.
The expression correctly applies the distributive property and the rules for combining like terms. Distributing −2-2 into (3x−4)(3x - 4) yields −6x+8-6x + 8. Combining the constants (8+88 + 8) gives 1616, and combining the variable terms (−6x+5x-6x + 5x) gives −x-x.

Step-by-Step Solution

1
Distribute the −2-2 to the terms inside the parentheses.
8−6x+8+5x8 - 6x + 8 + 5x
According to the order of operations (PEMDAS), the multiplication involved in the distributive property must be performed before addition and subtraction.
2
Group the like terms together.
(−6x+5x)+(8+8)(-6x + 5x) + (8 + 8)
Only terms with the same variable configuration (or constants) can be added or subtracted together.
3
Combine the like terms.
−x+16-x + 16
Adding the coefficients of the variable terms (−6+5=−1-6 + 5 = -1) and adding the constants (8+8=168 + 8 = 16) yields the fully simplified expression.

Key Concept

Simplifying Algebraic Expressions
Question 6Question →

Consider the algebraic expression:

−2a(3a−4b)−3b(a−2b)+5a2-2a(3a - 4b) - 3b(a - 2b) + 5a^2

Which of the following represents the correctly simplified form of this expression?

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Answer: −a2+5ab+6b2-a^2 + 5ab + 6b^2

Answer

The correctly simplified form is −a2+5ab+6b2-a^2 + 5ab + 6b^2.
The expression correctly applies the distributive property while carefully maintaining negative signs, and properly groups terms with identical variable combinations (a2a^2, abab, and b2b^2) to reach the final simplified form.

Step-by-Step Solution

1
Distribute −2a-2a into the first set of parentheses.
−6a2+8ab-6a^2 + 8ab
Applying the distributive property: (−2a×3a)+(−2a×−4b)(-2a \times 3a) + (-2a \times -4b). A negative term multiplied by a negative term results in a positive term.
2
Distribute −3b-3b into the second set of parentheses.
−3ab+6b2-3ab + 6b^2
Applying the distributive property: (−3b×a)+(−3b×−2b)(-3b \times a) + (-3b \times -2b). Again, the two negative factors produce a positive result.
3
Write out the full expanded expression by combining the distributed parts with the remaining term.
−6a2+8ab−3ab+6b2+5a2-6a^2 + 8ab - 3ab + 6b^2 + 5a^2
Setting up the expression to easily identify and group like terms together.
4
Group and combine the like terms.
−a2+5ab+6b2-a^2 + 5ab + 6b^2
Combining the a2a^2 terms (−6a2+5a2=−a2-6a^2 + 5a^2 = -a^2) and the abab terms (8ab−3ab=5ab8ab - 3ab = 5ab). The b2b^2 term remains unchanged.

Key Concept

Simplifying Expressions using the Distributive Property
Estimated Time:1m 0s
Question 7Question →

Which of the following is equivalent to the expression below?

2x(3x−4)−3[x(x−2)−4(2x+1)]+x22x(3x - 4) - 3[x(x - 2) - 4(2x + 1)] + x^2
Show answer & explanation

Answer: 4x2+22x+124x^2 + 22x + 12

Answer

The correct expression is 4x2+22x+124x^2 + 22x + 12.
The correct expression is found by following the order of operations: first expanding the terms inside the innermost parentheses, then combining like terms within the brackets, next distributing the negative multiplier to all terms in the brackets, and finally combining all remaining like terms.

Step-by-Step Solution

1
Expand the first part of the expression by distributing 2x2x.
2x(3x−4)=6x2−8x2x(3x - 4) = 6x^2 - 8x
Distribution is required to remove the initial parentheses.
2
Simplify the expression inside the square brackets by distributing and combining like terms.
x(x−2)−4(2x+1)=x2−2x−8x−4=x2−10x−4x(x - 2) - 4(2x + 1) = x^2 - 2x - 8x - 4 = x^2 - 10x - 4
According to the order of operations, simplify the innermost grouping symbols first.
3
Distribute the −3-3 across the simplified expression from the brackets.
−3(x2−10x−4)=−3x2+30x+12-3(x^2 - 10x - 4) = -3x^2 + 30x + 12
The negative sign and the constant must be multiplied by every term inside the brackets.
4
Substitute all parts back into the main expression and combine like terms.
(6x2−8x)+(−3x2+30x+12)+x2=(6x2−3x2+x2)+(−8x+30x)+12=4x2+22x+12(6x^2 - 8x) + (-3x^2 + 30x + 12) + x^2 = (6x^2 - 3x^2 + x^2) + (-8x + 30x) + 12 = 4x^2 + 22x + 12
Combining coefficients of terms with the same variables and exponents yields the final simplified polynomial.

Key Concept

Simplifying complex algebraic expressions by applying the distributive property, handling negative multipliers, and combining like terms.
Question 8Question →

Match each complex algebraic expression with its completely simplified polynomial form.

Click a left item, then click its matching right item

Items

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

Matches

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Answer

The first expression maps to −2x2−2xy-2x^2 - 2xy, the second expression maps to y2y^2, the third expression maps to −2xy+9y2-2xy + 9y^2, and the fourth expression maps to 2x2−4y22x^2 - 4y^2.
Each expression represents a distinct structural archetype for polynomial simplification. By systematically applying the order of operations—expanding binomials, distributing negative coefficients, and resolving nested grouping symbols from the inside out—all like terms can be properly combined to reach the correct canonical form.

Step-by-Step Solution

1
Simplify the first expression by resolving innermost parentheses, then brackets, then combining terms.
3x2−6xy−[2x2−4xy+3x2]3x^2 - 6xy - [2x^2 - 4xy + 3x^2] simplifies to 3x2−6xy−5x2+4xy3x^2 - 6xy - 5x^2 + 4xy, which yields −2x2−2xy-2x^2 - 2xy.
Following the order of operations from the innermost grouping symbols prevents sign errors when distributing negative coefficients.
2
Simplify the second expression by expanding the binomial square and distributing.
(4x2−4xy+y2)−(4x2−4xy)(4x^2 - 4xy + y^2) - (4x^2 - 4xy) simplifies directly to y2y^2.
Proper binomial expansion (a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2 avoids the common error of dropping the middle term, allowing opposite terms to cancel.
3
Simplify the third expression using the difference of squares formula, then distribute the negative.
−(x2−9y2)+x2−2xy-(x^2 - 9y^2) + x^2 - 2xy simplifies to −x2+9y2+x2−2xy-x^2 + 9y^2 + x^2 - 2xy, which reduces to −2xy+9y2-2xy + 9y^2.
Recognizing the difference of squares simplifies multiplication, and properly distributing the outer negative sign ensures the squared variable term has the correct positive sign.
4
Simplify the fourth expression by distributing within the brackets before multiplying by the outer coefficient.
4xy−2[−x2+2xy+2y2]4xy - 2[-x^2 + 2xy + 2y^2] simplifies to 4xy+2x2−4xy−4y2=2x2−4y24xy + 2x^2 - 4xy - 4y^2 = 2x^2 - 4y^2.
Combining like terms inside the bracket before multiplying by the outer constant significantly reduces the chance of sign and variable combination errors.

Key Concept

Simplifying Complex Multivariable Algebraic Expressions
Estimated Time:3m 0s
Question 9Question →

A student needs to simplify the expression 7k−2(k+4)+17k - 2(k + 4) + 1. What is the fully simplified form of this expression?

Show answer & explanation

Answer: 5k−75k - 7

Answer

The correct simplified expression is 5k−75k - 7.
The expression is simplified by first distributing the −2-2 into (k+4)(k + 4) to get −2k−8-2k - 8. Next, the like terms are combined: 7k−2k7k - 2k becomes 5k5k, and the constants −8-8 and 11 combine to −7-7. This results in the final simplified expression of 5k−75k - 7.

Step-by-Step Solution

1
Apply the distributive property to the term −2(k+4)-2(k + 4).
7k−2k−8+17k - 2k - 8 + 1
The multiplier −2-2 must be distributed to both kk and 44 inside the parentheses.
2
Group the like terms together.
(7k−2k)+(−8+1)(7k - 2k) + (-8 + 1)
Like terms (variables with the same exponent, and constants) are grouped so they can be combined.
3
Combine the grouped like terms to simplify the expression.
5k−75k - 7
Subtracting 2k2k from 7k7k gives 5k5k, and adding 11 to −8-8 gives −7-7.

Key Concept

Simplifying algebraic expressions by using the distributive property with negative multipliers and combining like terms.
Estimated Time:45s
Question 10Question →

An engineer is calculating the total weight supported by two structural pillars. The weight on the first pillar is given by the algebraic expression 5y−25y - 2, and the weight on the second pillar is given by 3y+63y + 6. What is the fully simplified algebraic expression for the total weight supported by both pillars combined?

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Answer: 8y + 4; 8y+4; 4 + 8y; 4+8y; 8y +4; 8y+ 4; 8y+ 4

Answer

8y + 4
The total weight is found by adding the two individual weight expressions. By identifying and grouping like terms, the variable terms are combined (5y+3y=8y5y + 3y = 8y) and the constant terms are combined (−2+6=4-2 + 6 = 4) to get the final simplified expression of 8y+48y + 4.

Step-by-Step Solution

1
Set up the addition of the two individual expressions
(5y−2)+(3y+6)(5y - 2) + (3y + 6)
To find the total weight, both individual weight expressions must be added together.
2
Group the like terms
(5y+3y)+(−2+6)(5y + 3y) + (-2 + 6)
Like terms (terms with the same variable, and constant terms) must be grouped to simplify the expression.
3
Combine the grouped terms
8y+48y + 4
Adding the coefficients of yy (5+3=85 + 3 = 8) and adding the constants (−2+6=4-2 + 6 = 4) yields the fully simplified form.

Key Concept

Combining Like Terms
Question 11Question →

Match each complex unsimplified algebraic expression with its correctly simplified equivalent.

Click a left item, then click its matching right item

Items

−3x[2y−(x−4y)]+2x(x−y)-3x[2y - (x - 4y)] + 2x(x - y)
4xy−2[x(3y−x)−y(x+2y)]4xy - 2[x(3y - x) - y(x + 2y)]
x2(y−2)−[xy(x−3)−x(2x+y)]−xyx^2(y - 2) - [xy(x - 3) - x(2x + y)] - xy
−(2x−3y)(x+y)+x(2x+y)−y(x−3y)-(2x - 3y)(x + y) + x(2x + y) - y(x - 3y)

Matches

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Answer

The correct pairings match each multi-step expression to its exact simplified polynomial counterpart: L1 matches R1, L2 matches R2, L3 matches R3, and L4 matches R4.
Each correct match is formed by strictly following the order of operations: clearing innermost groupings first, distributing outer multipliers across all contained terms, and combining structurally identical variable terms into standard canonical form.

Step-by-Step Solution

1
Simplify the first expression −3x[2y−(x−4y)]+2x(x−y)-3x[2y - (x - 4y)] + 2x(x - y).
The expression simplifies to 5x2−20xy5x^2 - 20xy.
By following the order of operations, innermost parentheses are cleared first, followed by distribution and combining like terms.
2
Simplify the second expression 4xy−2[x(3y−x)−y(x+2y)]4xy - 2[x(3y - x) - y(x + 2y)].
The expression simplifies to 2x2+4y22x^2 + 4y^2.
Expanding the terms inside the bracket yields −x2+2xy−2y2-x^2 + 2xy - 2y^2. Distributing the −2-2 flips the signs, and the xyxy terms successfully cancel.
3
Simplify the third expression x2(y−2)−[xy(x−3)−x(2x+y)]−xyx^2(y - 2) - [xy(x - 3) - x(2x + y)] - xy.
The expression simplifies to 3xy3xy.
Careful distribution of the negative sign across the simplified bracket (x2y−2x2−4xy)(x^2y - 2x^2 - 4xy) allows multiple high-degree terms to cancel out perfectly.
4
Simplify the fourth expression −(2x−3y)(x+y)+x(2x+y)−y(x−3y)-(2x - 3y)(x + y) + x(2x + y) - y(x - 3y).
The expression simplifies to xy+6y2xy + 6y^2.
FOILing the first binomial and keeping the negative sign outside ensures signs are correctly flipped before combining with the other distributed terms.

Key Concept

Simplifying complex algebraic expressions through precise application of the distributive property, order of operations (PEMDAS), and combination of like terms.
Estimated Time:4m 0s
Question 12Question →

What is the simplified form of the algebraic expression below?

2(4x+3)−x2(4x + 3) - x
Show answer & explanation

Answer: 7x+67x + 6

Answer

The correctly simplified expression is 7x+67x + 6.
To simplify the expression, first apply the distributive property by multiplying 2 by both terms inside the parentheses: 2(4x)+2(3)=8x+62(4x) + 2(3) = 8x + 6. Then, bring down the rest of the expression: 8x+6−x8x + 6 - x. Finally, combine the like terms involving xx: 8x−x=7x8x - x = 7x. The simplified result is 7x+67x + 6.

Step-by-Step Solution

1
Distribute the 2 to both terms inside the parentheses.
8x+6−x8x + 6 - x
The distributive property requires multiplying the outside term by every term inside the parentheses.
2
Combine the like terms (8x8x and −x-x).
7x+67x + 6
Only terms with the same variable and exponent can be combined.

Key Concept

Simplifying Algebraic Expressions (Distributive Property and Combining Like Terms)
Question 13Question →

Consider the algebraic expression:

3a(2a−5b)−2b(a+4b)+7ab3a(2a - 5b) - 2b(a + 4b) + 7ab

When completely simplified, what is the resulting polynomial?

Show answer & explanation

Answer: 6a2−10ab−8b26a^2 - 10ab - 8b^2

Answer

The simplified polynomial is 6a2−10ab−8b26a^2 - 10ab - 8b^2.
The correct answer is found by first distributing the outer terms to eliminate parentheses, carefully observing negative signs. Distributing 3a3a gives 6a2−15ab6a^2 - 15ab. Distributing −2b-2b gives −2ab−8b2-2ab - 8b^2. Adding the +7ab+7ab at the end produces the expanded expression 6a2−15ab−2ab−8b2+7ab6a^2 - 15ab - 2ab - 8b^2 + 7ab. Combining the like terms (−15ab−2ab+7ab-15ab - 2ab + 7ab) yields −10ab-10ab, resulting in the final expression 6a2−10ab−8b26a^2 - 10ab - 8b^2.

Step-by-Step Solution

1
Distribute 3a3a to the terms inside the first set of parentheses.
3a(2a)−3a(5b)=6a2−15ab3a(2a) - 3a(5b) = 6a^2 - 15ab
Applying the distributive property to eliminate the first set of parentheses.
2
Distribute −2b-2b to the terms inside the second set of parentheses.
−2b(a)−2b(4b)=−2ab−8b2-2b(a) - 2b(4b) = -2ab - 8b^2
Applying the distributive property, ensuring the negative sign is multiplied into both terms.
3
Rewrite the full expression with the expanded terms.
6a2−15ab−2ab−8b2+7ab6a^2 - 15ab - 2ab - 8b^2 + 7ab
Putting all terms together to prepare for combining like terms.
4
Identify and combine the like terms (the abab terms).
−15ab−2ab+7ab=−10ab-15ab - 2ab + 7ab = -10ab
Only terms with the exact same variable combinations can be added or subtracted.
5
Construct the final polynomial.
6a2−10ab−8b26a^2 - 10ab - 8b^2
The a2a^2 and b2b^2 terms cannot be combined with anything else.

Key Concept

Simplifying algebraic expressions using the distributive property and combining like terms.
Question 14Question →

Which of the following is the completely simplified form of the algebraic expression below?

4a(2a−3b)−[3b(a−4b)−2a(a+b)]4a(2a - 3b) - [3b(a - 4b) - 2a(a + b)]
Show answer & explanation

Answer: 10a2−13ab+12b210a^2 - 13ab + 12b^2

Answer

The simplified expression is 10a2−13ab+12b210a^2 - 13ab + 12b^2.
The correct answer is derived by first expanding the inner products (3ab−12b23ab - 12b^2 and −2a2−2ab-2a^2 - 2ab), combining them to get ab−12b2−2a2ab - 12b^2 - 2a^2, and then subtracting this entire grouped result from the expanded outer product (8a2−12ab8a^2 - 12ab). Properly distributing the negative sign to all terms from the bracket yields 10a2−13ab+12b210a^2 - 13ab + 12b^2.

Step-by-Step Solution

1
Distribute the terms inside the square brackets.
3b(a−4b)=3ab−12b23b(a - 4b) = 3ab - 12b^2 and −2a(a+b)=−2a2−2ab-2a(a + b) = -2a^2 - 2ab
According to the order of operations, simplify the innermost expressions first.
2
Combine like terms inside the brackets.
[3ab−12b2−2a2−2ab]=[ab−12b2−2a2][3ab - 12b^2 - 2a^2 - 2ab] = [ab - 12b^2 - 2a^2]
Simplifying the bracket's contents reduces complexity before distributing the negative sign.
3
Distribute the terms outside the brackets.
4a(2a−3b)=8a2−12ab4a(2a - 3b) = 8a^2 - 12ab
Expand the remaining multiplication.
4
Distribute the negative sign to the terms from the bracket.
−(ab−12b2−2a2)=−ab+12b2+2a2-(ab - 12b^2 - 2a^2) = -ab + 12b^2 + 2a^2
Subtracting a polynomial requires reversing the sign of every term inside.
5
Combine all remaining like terms.
8a2−12ab−ab+12b2+2a2=10a2−13ab+12b28a^2 - 12ab - ab + 12b^2 + 2a^2 = 10a^2 - 13ab + 12b^2
Group terms with identical variable parts to find the final simplified form.

Key Concept

Simplifying complex expressions by adhering strictly to the order of operations, accurately distributing terms and negative signs, and properly combining like terms.
Question 15Question →

Match each unsimplified algebraic expression on the left with its correctly simplified form on the right.

Click a left item, then click its matching right item

Items

4m+7m−24m + 7m - 2
3(2m+5)3(2m + 5)
9m−(4m−3)9m - (4m - 3)
m(2m−1)m(2m - 1)

Matches

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Answer

The correct matches pair each unsimplified expression with its corresponding simplified form: 4m + 7m - 2 pairs with 11m - 2; 3(2m + 5) pairs with 6m + 15; 9m - (4m - 3) pairs with 5m + 3; and m(2m - 1) pairs with 2m^2 - m.
Each unsimplified expression is paired with its mathematically equivalent simplified form by correctly applying standard algebraic rules: combining like terms and utilizing the distributive property.

Step-by-Step Solution

1
Simplify the first expression by identifying and combining like terms.
4m+7m−2=11m−24m + 7m - 2 = 11m - 2
The terms 4m and 7m have the exact same variable part and can be added together.
2
Simplify the second expression by applying the distributive property.
3(2m+5)=6m+153(2m + 5) = 6m + 15
The outside factor of 3 must be multiplied by every term contained inside the parentheses.
3
Simplify the third expression by distributing the negative sign, then combining the resulting like terms.
9m−4m+3=5m+39m - 4m + 3 = 5m + 3
Subtracting a grouped polynomial requires reversing the sign of each term within the group.
4
Simplify the fourth expression by multiplying the variable through the parentheses.
m(2m−1)=2m2−mm(2m - 1) = 2m^2 - m
Multiplying a variable by itself increases its exponent based on the laws of exponents.

Key Concept

Simplifying Algebraic Expressions
Question 16Question →

Pair each unsimplified algebraic expression with its correctly simplified polynomial equivalent in standard form.

Click a left item, then click its matching right item

Items

3p(2p−q)−2(p2+4pq)3p(2p - q) - 2(p^2 + 4pq)
−4(p2−2pq)+2p(p+3q)-4(p^2 - 2pq) + 2p(p + 3q)
p(5p+2q)−(p2−3pq)p(5p + 2q) - (p^2 - 3pq)
2(3p2−pq)−p(8p−5q)2(3p^2 - pq) - p(8p - 5q)

Matches

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Answer

The expressions simplify as follows: the first expression simplifies to 4p² - 11pq; the second to -2p² + 14pq; the third to 4p² + 5pq; and the fourth to -2p² + 3pq.
Each expression is correctly simplified by first distributing the outer multipliers to every term inside their respective parentheses, paying close attention to negative signs. Once expanded, like terms (terms containing p² and terms containing pq) are combined by adding or subtracting their coefficients.

Step-by-Step Solution

1
Simplify the first expression: 3p(2p - q) - 2(p² + 4pq)
6p² - 3pq - 2p² - 8pq = 4p² - 11pq
Distribution of the outside terms into the parentheses, followed by grouping like terms.
2
Simplify the second expression: -4(p² - 2pq) + 2p(p + 3q)
-4p² + 8pq + 2p² + 6pq = -2p² + 14pq
Carefully distribute the negative constant (-4), changing the inner sign to positive 8pq, then combine like terms.
3
Simplify the third expression: p(5p + 2q) - (p² - 3pq)
5p² + 2pq - p² + 3pq = 4p² + 5pq
Distribute the variable p and distribute the negative sign to flip the inner signs, then collect like terms.
4
Simplify the fourth expression: 2(3p² - pq) - p(8p - 5q)
6p² - 2pq - 8p² + 5pq = -2p² + 3pq
Distribute the 2 and the -p, ensuring the multiplication of -p and -5q results in +5pq, then combine.

Key Concept

Simplifying Algebraic Expressions
Question 17Question →

Evaluate the following multivariable algebraic expressions. Pair each original expression with its simplified form in standard notation.

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Items

−3a(2a−b)−(a−2b)(2a+b)-3a(2a - b) - (a - 2b)(2a + b)
2ab(a−3b)−a(2ab−5b2)2ab(a - 3b) - a(2ab - 5b^2)
4(a2−ab+b2)−(2a−b)24(a^2 - ab + b^2) - (2a - b)^2
a2(2b−1)−b(a2−2a)+(a−b)2−b2a^2(2b - 1) - b(a^2 - 2a) + (a - b)^2 - b^2

Matches

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Answer

Each unsimplified expression matches exactly one standard form polynomial upon successful expansion and collection of like terms.
Each pairing is correct because the unsimplified expressions rigidly evaluate to the exact standard forms provided when algebraic expansion and simplification rules are followed without computational errors.

Step-by-Step Solution

1
Simplify the first expression by expanding and subtracting the grouped terms.
−3a(2a−b)−(a−2b)(2a+b)-3a(2a - b) - (a - 2b)(2a + b) simplifies to −8a2+6ab+2b2-8a^2 + 6ab + 2b^2.
Applying the distributive property and FOIL method accurately, followed by subtracting the terms (which reverses their signs).
2
Simplify the second expression by applying distribution to both halves.
2ab(a−3b)−a(2ab−5b2)2ab(a - 3b) - a(2ab - 5b^2) simplifies to −ab2-ab^2.
Distributing the multipliers across the parentheses isolates terms that naturally cancel or combine.
3
Simplify the third expression by expanding the perfect square.
4(a2−ab+b2)−(2a−b)24(a^2 - ab + b^2) - (2a - b)^2 simplifies to 3b23b^2.
The expansion of (2a−b)2(2a - b)^2 creates terms that precisely offset the a2a^2 and abab terms in the distributed trinomial.
4
Simplify the fourth expression by expanding all grouped and squared segments.
a2(2b−1)−b(a2−2a)+(a−b)2−b2a^2(2b - 1) - b(a^2 - 2a) + (a - b)^2 - b^2 simplifies to a2ba^2b.
Combining the resulting elements reveals that nearly all terms possess additive inverses, leaving only the combination of 2a2b−a2b2a^2b - a^2b.

Key Concept

Expanding polynomials, applying the distributive property with negative signs, and collecting like variables.
Question 18Question →

Match each unsimplified algebraic expression on the left with its correctly simplified polynomial equivalent on the right.

Click a left item, then click its matching right item

Items

3x(2x−4)−2(x2−5x)3x(2x - 4) - 2(x^2 - 5x)
5(x2−3x+2)−4x(x−2)5(x^2 - 3x + 2) - 4x(x - 2)
−2x(3x+1)+x(5x−4)-2x(3x + 1) + x(5x - 4)
4(2x2−x)−3(x2+2x−1)4(2x^2 - x) - 3(x^2 + 2x - 1)

Matches

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Answer

The correct pairings link each unsimplified expression with its resulting simplified polynomial after applying the distributive property and combining like terms.
Each unsimplified expression is accurately paired with its simplified equivalent by first applying the distributive property to eliminate all parentheses, rigorously managing negative multipliers, and finally grouping and combining all like terms according to their respective exponents.

Step-by-Step Solution

1
Expand and simplify the first expression.
6x2−12x−2x2+10x6x^2 - 12x - 2x^2 + 10x, which simplifies to 4x2−2x4x^2 - 2x.
Applying the distributive property and combining the x2x^2 terms and the xx terms.
2
Expand and simplify the second expression.
5x2−15x+10−4x2+8x5x^2 - 15x + 10 - 4x^2 + 8x, which simplifies to x2−7x+10x^2 - 7x + 10.
Distributing the constants and variables across parentheses, paying special attention to the negative signs.
3
Expand and simplify the third expression.
−6x2−2x+5x2−4x-6x^2 - 2x + 5x^2 - 4x, which simplifies to −x2−6x-x^2 - 6x.
Multiplying each term inside the parentheses by the outside factor and combining the resulting like terms.
4
Expand and simplify the fourth expression.
8x2−4x−3x2−6x+38x^2 - 4x - 3x^2 - 6x + 3, which simplifies to 5x2−10x+35x^2 - 10x + 3.
Distributing the −3-3 correctly reverses the signs of all terms within the second polynomial.

Key Concept

Simplifying Algebraic Expressions
Question 19Question →
Which of the following expressions is equivalent to 15−3(2k+4)15 - 3(2k + 4) after applying the proper order of operations and combining like terms?
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Answer: −6k+3-6k + 3

Answer

−6k+3-6k + 3
The correct answer is found by first distributing the -3 to both the 2k and the 4, yielding -6k - 12. Then, combining the constant terms 15 and -12 results in a final, standard form simplified expression of -6k + 3.

Step-by-Step Solution

1
Distribute the -3 to both terms inside the parentheses.
15−6k−1215 - 6k - 12
According to the order of operations, multiplication (distribution) must be performed before subtraction.
2
Identify and group the like terms.
−6k+(15−12)-6k + (15 - 12)
Constant numbers are like terms and must be combined to fully simplify the expression.
3
Perform the arithmetic to combine the constants.
−6k+3-6k + 3
Subtracting 12 from 15 yields the final constant value in the expression.

Key Concept

Distributive property and combining like terms
Question 20Question →

Simplify the following algebraic expression:

12−3(2y−4)12 - 3(2y - 4)

Which expression represents the fully simplified result?

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Answer: −6y+24-6y + 24

Answer

The correct simplified expression is −6y+24-6y + 24.
The expression is correctly simplified by first distributing the −3-3 across the terms in the parentheses, which results in −6y+12-6y + 12. Adding the initial 1212 to this new constant gives a final expression of −6y+24-6y + 24.

Step-by-Step Solution

1
Distribute the −3-3 to both terms inside the parentheses.
12−6y+1212 - 6y + 12
The distributive property requires multiplying the term immediately outside the parentheses by each term inside.
2
Combine the constant terms.
−6y+24-6y + 24
Like terms (the constants 1212 and 1212) must be added together to fully simplify the expression.

Key Concept

Distributive Property and Combining Like Terms
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