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Simplifying Algebraic Expressions

33 questions

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Question 21Question →

An algebraic expression is given below:

−2p(3q−p)−[4pq−p(2p+5q)]-2p(3q - p) - [4pq - p(2p + 5q)]

When fully simplified, which of the following represents the equivalent polynomial?

Show answer & explanation

Answer: 4p2−5pq4p^2 - 5pq

Answer

4p2−5pq4p^2 - 5pq
The expression simplifies correctly by first distributing the multipliers: −2p(3q−p)=−6pq+2p2-2p(3q - p) = -6pq + 2p^2 and −p(2p+5q)=−2p2−5pq-p(2p + 5q) = -2p^2 - 5pq. The bracket becomes [4pq−2p2−5pq]=[−pq−2p2][4pq - 2p^2 - 5pq] = [-pq - 2p^2]. Distributing the negative sign outside the bracket gives +pq+2p2+pq + 2p^2. Finally, combining all terms: (−6pq+2p2)+(pq+2p2)=4p2−5pq(-6pq + 2p^2) + (pq + 2p^2) = 4p^2 - 5pq.

Step-by-Step Solution

1
Distribute the outer multiplier −2p-2p into the first binomial.
−6pq+2p2-6pq + 2p^2
Applying the distributive property: −2p(3q)=−6pq-2p(3q) = -6pq and −2p(−p)=+2p2-2p(-p) = +2p^2.
2
Distribute −p-p into the binomial within the square brackets.
4pq−2p2−5pq4pq - 2p^2 - 5pq
Applying the distributive property inside the grouping: −p(2p)=−2p2-p(2p) = -2p^2 and −p(5q)=−5pq-p(5q) = -5pq.
3
Combine like terms inside the square brackets.
−pq−2p2-pq - 2p^2
Combining the like terms 4pq4pq and −5pq-5pq yields −pq-pq.
4
Distribute the negative sign outside the square brackets to the simplified contents.
+pq+2p2+pq + 2p^2
Subtracting a quantity is equivalent to adding its opposite: −(−pq−2p2)=+pq+2p2-(-pq - 2p^2) = +pq + 2p^2.
5
Combine all remaining terms from the entire expression.
2p2−6pq+pq+2p2=4p2−5pq2p^2 - 6pq + pq + 2p^2 = 4p^2 - 5pq
Grouping like terms: (2p2+2p2)+(−6pq+pq)=4p2−5pq(2p^2 + 2p^2) + (-6pq + pq) = 4p^2 - 5pq.

Key Concept

Simplifying expressions involves correctly applying the distributive property, especially with negative multipliers, and combining like terms.
Question 22Question →

A mathematical model for a fluid dynamics problem reduces to the following algebraic expression:

−2xy(3x−4y)−[5x2y−2xy(x+3y)] -2xy(3x - 4y) - [5x^2y - 2xy(x + 3y)]

When completely simplified, which of the following represents the equivalent standard form?

Show answer & explanation

Answer: −9x2y+14xy2-9x^2y + 14xy^2

Answer

The correct expression is −9x2y+14xy2-9x^2y + 14xy^2.
The correct expression is derived by applying the order of operations and the distributive property. First, evaluate the terms inside the brackets to get 3x2y−6xy23x^2y - 6xy^2. Then expand the initial term to −6x2y+8xy2-6x^2y + 8xy^2. Subtracting the bracket flips the inner signs, yielding −6x2y+8xy2−3x2y+6xy2-6x^2y + 8xy^2 - 3x^2y + 6xy^2. Combining like terms results in −9x2y+14xy2-9x^2y + 14xy^2.

Step-by-Step Solution

1
Expand the inner terms inside the brackets by distributing −2xy-2xy to (x+3y)(x + 3y).
5x2y−2x2y−6xy25x^2y - 2x^2y - 6xy^2
Following the order of operations, simplify grouping symbols first.
2
Combine like terms within the brackets.
3x2y−6xy23x^2y - 6xy^2
To minimize the expression before subtracting it from the leading terms.
3
Distribute −2xy-2xy to the terms in the first parenthesis (3x−4y)(3x - 4y).
−6x2y+8xy2-6x^2y + 8xy^2
To expand the first part of the expression.
4
Subtract the simplified bracketed expression from the expanded first part.
−6x2y+8xy2−(3x2y−6xy2)=−6x2y+8xy2−3x2y+6xy2-6x^2y + 8xy^2 - (3x^2y - 6xy^2) = -6x^2y + 8xy^2 - 3x^2y + 6xy^2
To remove the remaining grouping symbols and properly distribute the negative sign.
5
Combine the remaining like terms to reach the final form.
−9x2y+14xy2-9x^2y + 14xy^2
To express the polynomial in its simplest standard form.

Key Concept

Simplifying complex expressions using the distributive property, managing negative multipliers, and working with nested grouping symbols.
Question 23Question →

Determine the completely simplified form of the following expression:

3(2a+4)−5a3(2a + 4) - 5a
Show answer & explanation

Answer: a + 12; a+12; 12 + a; 12+a; 1a + 12; 1a+12

Answer

a + 12
Applying the distributive property to 3(2a+4)3(2a + 4) yields 6a+126a + 12. The expression becomes 6a+12−5a6a + 12 - 5a. Subtracting 5a5a from 6a6a leaves aa. The constant 1212 cannot be combined with aa, making the final simplified expression a+12a + 12.

Step-by-Step Solution

1
Use the distributive property to multiply 3 by each term inside the parentheses.
6a+12−5a6a + 12 - 5a
Expanding the parentheses is the necessary first step to access the terms for combination.
2
Group and combine the like terms (6a6a and −5a-5a).
a+12a + 12
Subtracting 5a5a from 6a6a leaves 1a1a (or aa), while the constant 1212 remains unchanged.

Key Concept

Applying the distributive property and combining like terms.
Estimated Time:45s
Question 24Question →

Simplify the following algebraic expression completely:

3(4m−2n)−2(3m+5n)−(m−12n)3(4m - 2n) - 2(3m + 5n) - (m - 12n)
Show answer & explanation

Answer: 5m - 4n; 5m-4n; -4n + 5m; -4n+5m

Answer

5m - 4n
To simplify the expression, first apply the distributive property to eliminate the parentheses, taking care to distribute the negative signs properly: 12m−6n−6m−10n−m+12n12m - 6n - 6m - 10n - m + 12n. Then, combine the like terms for mm (12m−6m−m=5m12m - 6m - m = 5m) and the like terms for nn (−6n−10n+12n=−4n-6n - 10n + 12n = -4n) to reach the final answer of 5m−4n5m - 4n.

Step-by-Step Solution

1
Distribute the coefficients to the terms inside each set of parentheses.
12m−6n−6m−10n−m+12n12m - 6n - 6m - 10n - m + 12n
The distributive property must be applied first to remove parentheses before like terms can be combined.
2
Group the like terms for variables mm and nn.
(12m−6m−m)+(−6n−10n+12n)(12m - 6m - m) + (-6n - 10n + 12n)
Grouping like terms organizes the expression to easily combine their coefficients.
3
Combine the coefficients of the grouped terms.
5m−4n5m - 4n
Evaluating the addition and subtraction of the coefficients yields the fully simplified expression.

Key Concept

Simplifying Algebraic Expressions
Question 25Question →

What is the result of fully expanding and simplifying the given algebraic expression?

−4k(2k−3m)+5k(k−2m)−3k2-4k(2k - 3m) + 5k(k - 2m) - 3k^2
Show answer & explanation

Answer: −6k2+2km-6k^2 + 2km

Answer

The correct simplified expression is −6k2+2km-6k^2 + 2km.
The correct answer accurately expands the expression by distributing −4k-4k to get −8k2+12km-8k^2 + 12km, and 5k5k to get 5k2−10km5k^2 - 10km. Grouping the k2k^2 terms (−8+5−3=−6-8 + 5 - 3 = -6) and the kmkm terms (12−10=212 - 10 = 2) yields −6k2+2km-6k^2 + 2km.

Step-by-Step Solution

1
Distribute −4k-4k across the terms inside the first set of parentheses: (2k−3m)(2k - 3m).
−8k2+12km-8k^2 + 12km
To eliminate the first grouping symbol, being careful to multiply the negative signs properly (−4k×−3m=+12km-4k \times -3m = +12km).
2
Distribute 5k5k across the terms inside the second set of parentheses: (k−2m)(k - 2m).
5k2−10km5k^2 - 10km
To eliminate the second grouping symbol.
3
Write out the complete expanded expression by combining the results and the remaining −3k2-3k^2 term.
−8k2+12km+5k2−10km−3k2-8k^2 + 12km + 5k^2 - 10km - 3k^2
This shows all individual terms ready for grouping.
4
Identify and combine the like terms: the k2k^2 terms (−8k2-8k^2, 5k25k^2, −3k2-3k^2) and the kmkm terms (12km12km, −10km-10km).
−6k2+2km-6k^2 + 2km
Combining like coefficients (−8+5−3=−6-8 + 5 - 3 = -6 and 12−10=212 - 10 = 2) provides the final simplified expression in standard form.

Key Concept

Applying the distributive property with negative numbers and combining multivariable like terms.
Question 26Question →

Match each unsimplified algebraic expression with its correctly simplified equivalent.

Click a left item, then click its matching right item

Items

2(3y+4)2(3y + 4)
5y+2y−35y + 2y - 3
−3(y−2)-3(y - 2)
4y+2z−y+3z4y + 2z - y + 3z

Matches

Show answer & explanation

Answer

The correct pairings map each unsimplified expression to its distributed and combined form.
Each expression on the left is matched with its simplified form on the right by accurately applying the distributive property and combining like terms.

Step-by-Step Solution

1
Simplify the first expression: 2(3y + 4).
6y + 8
Multiply the outer term 2 by both terms inside the parentheses (3y and 4).
2
Simplify the second expression: 5y + 2y - 3.
7y - 3
Add the coefficients of the like terms containing y (5 and 2).
3
Simplify the third expression: -3(y - 2).
-3y + 6
Distribute the -3, noting that a negative multiplied by a negative results in a positive (+6).
4
Simplify the fourth expression: 4y + 2z - y + 3z.
3y + 5z
Group and add the coefficients of the y terms together, and the z terms together.

Key Concept

Simplifying Algebraic Expressions
Question 27Question →

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

Click a left item, then click its matching right item

Items

3(2a−b)−4(a+2b)3(2a - b) - 4(a + 2b)
−2(a−3b)+5(2a−b)-2(a - 3b) + 5(2a - b)
a(3−2b)−2(a−ab)a(3 - 2b) - 2(a - ab)
4a−[2b−3(a−b)]4a - [2b - 3(a - b)]

Matches

Show answer & explanation

Answer

The unsimplified expressions must be correctly expanded using the distributive property, observing proper handling of negative multipliers and nested grouping symbols, and then reduced by combining like terms.
Each expression correctly matches its fully reduced polynomial form. The reductions require systematic application of the distributive property, careful tracking of positive and negative coefficients, and the grouping and addition of like variables.

Step-by-Step Solution

1
Simplify the first expression: 3(2a−b)−4(a+2b)3(2a - b) - 4(a + 2b)
6a−3b−4a−8b=2a−11b6a - 3b - 4a - 8b = 2a - 11b
Multiply each term inside the parentheses by the outside factor, then group like terms (6a−4a6a - 4a and −3b−8b-3b - 8b).
2
Simplify the second expression: −2(a−3b)+5(2a−b)-2(a - 3b) + 5(2a - b)
−2a+6b+10a−5b=8a+b-2a + 6b + 10a - 5b = 8a + b
Properly distribute the negative sign in the first term (−2×−3b=6b-2 \times -3b = 6b), then combine like terms.
3
Simplify the third expression: a(3−2b)−2(a−ab)a(3 - 2b) - 2(a - ab)
3a−2ab−2a+2ab=a3a - 2ab - 2a + 2ab = a
Distribute the variables and constants, recognizing that −2ab-2ab and +2ab+2ab are additive inverses and cancel to zero.
4
Simplify the fourth expression: 4a−[2b−3(a−b)]4a - [2b - 3(a - b)]
4a−[2b−3a+3b]=4a−[5b−3a]=4a−5b+3a=7a−5b4a - [2b - 3a + 3b] = 4a - [5b - 3a] = 4a - 5b + 3a = 7a - 5b
Work from the innermost parentheses outward. Distribute −3-3, combine like terms inside the bracket, and distribute the outer negative sign.

Key Concept

Simplifying Algebraic Expressions
Estimated Time:2m 0s
Question 28Question →

Simplify the following algebraic expression:

8x−2[x−3(2x−y)]8x - 2[x - 3(2x - y)]

Which of the following represents the completely simplified form?

Show answer & explanation

Answer: 18x−6y18x - 6y

Answer

The completely simplified form is 18x−6y18x - 6y.
The expression is fully simplified by applying the distributive property iteratively, starting from the innermost parentheses, carefully tracking all negative signs, and finally combining like terms to arrive at 18x−6y18x - 6y.

Step-by-Step Solution

1
Distribute the -3 into the innermost parentheses.
−3(2x−y)=−6x+3y-3(2x - y) = -6x + 3y
According to the order of operations, innermost grouping symbols must be evaluated first. Distributing a negative number flips the signs of the terms inside.
2
Substitute this back and combine like terms inside the brackets.
x−6x+3y=−5x+3yx - 6x + 3y = -5x + 3y
Combining the like variable terms simplifies the expression inside the brackets before moving to the next operation.
3
Distribute the -2 across the newly simplified terms inside the bracket.
−2(−5x+3y)=10x−6y-2(-5x + 3y) = 10x - 6y
The distributive property applies the outside multiplier to every term within the brackets.
4
Combine this result with the remaining term outside the brackets.
8x+10x−6y=18x−6y8x + 10x - 6y = 18x - 6y
Adding the like x terms yields the final, completely simplified algebraic expression.

Key Concept

Simplifying Nested Algebraic Expressions
Question 29Question →

Simplify the following algebraic expression completely by distributing and combining like terms:

3a(2a−4b)−2b(a−5b)3a(2a - 4b) - 2b(a - 5b)
Show answer & explanation

Answer: 6a^2 - 14ab + 10b^2; 6a^2-14ab+10b^2; 10b^2 - 14ab + 6a^2; 6a^2+10b^2-14ab; 6a^2 - 14ba + 10b^2

Answer

6a^2 - 14ab + 10b^2
The expression is correctly simplified by first applying the distributive property to both parts: 3a(2a−4b)3a(2a - 4b) becomes 6a2−12ab6a^2 - 12ab, and −2b(a−5b)-2b(a - 5b) becomes −2ab+10b2-2ab + 10b^2. Next, the like terms (−12ab-12ab and −2ab-2ab) are combined to form −14ab-14ab. The resulting expression in standard form is 6a2−14ab+10b26a^2 - 14ab + 10b^2.

Step-by-Step Solution

1
Distribute 3a3a across the terms in the first parenthesis (2a−4b)(2a - 4b).
6a2−12ab6a^2 - 12ab
Applying the distributive property to eliminate the first set of parentheses.
2
Distribute −2b-2b across the terms in the second parenthesis (a−5b)(a - 5b).
−2ab+10b2-2ab + 10b^2
Multiplying by a negative requires changing the signs. Specifically, (−2b)×(−5b)=+10b2(-2b) \times (-5b) = +10b^2.
3
Write out the full expanded expression and identify like terms.
6a2−12ab−2ab+10b26a^2 - 12ab - 2ab + 10b^2
Preparing to combine terms with identical variable parts.
4
Combine the like terms (−12ab-12ab and −2ab-2ab).
6a2−14ab+10b26a^2 - 14ab + 10b^2
−12−2=−14-12 - 2 = -14, resulting in the final simplified standard form.

Key Concept

Distributive Property and Combining Like Terms with Multiple Variables
Question 30Question →

Consider the polynomial expression provided below:

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

Which of the following represents its fully simplified equivalent?

Show answer & explanation

Answer: 15x−1415x - 14

Answer

The completely simplified equivalent expression is 15x−1415x - 14.
The expression correctly expands to 2x2+5x−122x^2 + 5x - 12 on the left and −2x2+10x−2-2x^2 + 10x - 2 on the right. When adding these together, the x2x^2 terms cancel out, the xx terms sum to 15x15x, and the constants sum to −14-14, leaving 15x−1415x - 14.

Step-by-Step Solution

1
Expand (2x−3)(x+4)(2x - 3)(x + 4) using the distributive property.
2x2+8x−3x−122x^2 + 8x - 3x - 12, which simplifies to 2x2+5x−122x^2 + 5x - 12.
To eliminate the first set of parentheses by multiplying each term in the first binomial by each term in the second binomial.
2
Distribute −2-2 to each term inside the trinomial (x2−5x+1)(x^2 - 5x + 1).
−2x2+10x−2-2x^2 + 10x - 2.
To eliminate the second set of parentheses, carefully applying the negative sign to all inner terms.
3
Combine the expanded expressions.
(2x2+5x−12)+(−2x2+10x−2)(2x^2 + 5x - 12) + (-2x^2 + 10x - 2).
To prepare the entire expression for the collection of like terms.
4
Group and combine like terms (x2x^2 terms, xx terms, and constants).
(2x2−2x2)+(5x+10x)+(−12−2)=0x2+15x−14=15x−14(2x^2 - 2x^2) + (5x + 10x) + (-12 - 2) = 0x^2 + 15x - 14 = 15x - 14.
To achieve the final simplified standard form.

Key Concept

Expanding polynomials, distributing negative coefficients, and combining like terms.
Question 31Question →

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

Click a left item, then click its matching right item

Items

5x−3(2x−4)5x - 3(2x - 4)
3(2m−5n)−2(m−3n)3(2m - 5n) - 2(m - 3n)
4x(x−2)−x2+5x4x(x - 2) - x^2 + 5x
7y−3[y−2(y+1)]7y - 3[y - 2(y + 1)]

Matches

Show answer & explanation

Answer

The correctly simplified expressions are: 5x−3(2x−4)5x - 3(2x - 4) matches with −x+12-x + 12; 3(2m−5n)−2(m−3n)3(2m - 5n) - 2(m - 3n) matches with 4m−9n4m - 9n; 4x(x−2)−x2+5x4x(x - 2) - x^2 + 5x matches with 3x2−3x3x^2 - 3x; and 7y−3[y−2(y+1)]7y - 3[y - 2(y + 1)] matches with 10y+610y + 6.
Each unsimplified expression must be correctly expanded using the distributive property, ensuring that negative multipliers are applied to every term inside the parentheses. After expanding, grouping and adding the coefficients of like terms produces the single correct simplified equivalent provided in the match list.

Step-by-Step Solution

1
Distribute any coefficients or negative signs immediately outside of parentheses or brackets.
Terms are expanded (e.g., −3(2x−4)-3(2x - 4) becomes −6x+12-6x + 12).
Distribution removes grouping symbols, which is required before combining terms.
2
For expressions with nested grouping symbols, simplify from the innermost parentheses outward.
Complex expressions like 7y−3[y−2(y+1)]7y - 3[y - 2(y + 1)] simplify step-by-step to 7y+3y+67y + 3y + 6.
Following the order of operations ensures accurate distribution of multipliers.
3
Group and combine like terms (terms with the exact same variable and exponent).
The final canonical form is reached for each expression.
Combining like terms fully simplifies the polynomial into standard form.

Key Concept

Simplifying Algebraic Expressions
Estimated Time:1m 30s
Question 32Question →

What is the completely simplified form of the algebraic expression 7x−4(2x−3y)+5y7x - 4(2x - 3y) + 5y?

Show answer & explanation

Answer: −x+17y-x + 17y

Answer

−x+17y-x + 17y
To completely simplify the expression, you must first distribute the −4-4 across the terms in the parentheses. Multiplying −4-4 by 2x2x gives −8x-8x, and multiplying −4-4 by −3y-3y gives +12y+12y (since a negative times a negative is a positive). Substituting this back gives 7x−8x+12y+5y7x - 8x + 12y + 5y. Finally, group the like variables: 7x−8x=−x7x - 8x = -x, and 12y+5y=17y12y + 5y = 17y. Combining these gives −x+17y-x + 17y.

Step-by-Step Solution

1
Distribute the −4-4 to both terms inside the parentheses (2x−3y)(2x - 3y).
−8x+12y-8x + 12y
The distributive property must be applied first to eliminate the parentheses, paying careful attention to the multiplication of two negative numbers (−4-4 and −3y-3y).
2
Rewrite the full expression incorporating the expanded terms.
7x−8x+12y+5y7x - 8x + 12y + 5y
This sets up the expression so that all individual terms are clearly visible for grouping.
3
Combine the like terms for both xx and yy.
−x+17y-x + 17y
Adding the coefficients of identical variables (7x−8x=−x7x - 8x = -x and 12y+5y=17y12y + 5y = 17y) provides the expression in its final standard form.

Key Concept

Simplifying Algebraic Expressions via Distributive Property and Like Terms
Question 33Question →

Which of the following is equivalent to the algebraic expression below after it has been fully simplified?

4k(k−3m)−(2k2−5km+m2)4k(k - 3m) - (2k^2 - 5km + m^2)
Show answer & explanation

Answer: 2k2−7km−m22k^2 - 7km - m^2

Answer

The fully simplified expression is 2k2−7km−m22k^2 - 7km - m^2.
The correct expression is derived by first multiplying 4k4k by both terms in (k−3m)(k - 3m) to get 4k2−12km4k^2 - 12km. Next, distributing the negative sign through the trinomial changes the signs to yield −2k2+5km−m2-2k^2 + 5km - m^2. Finally, grouping and combining like terms yields 4k2−2k2=2k24k^2 - 2k^2 = 2k^2 and −12km+5km=−7km-12km + 5km = -7km. The −m2-m^2 term has no like terms to combine with, leaving it unchanged. The final simplified form is 2k2−7km−m22k^2 - 7km - m^2.

Step-by-Step Solution

1
Distribute 4k4k into the binomial (k−3m)(k - 3m).
4k2−12km4k^2 - 12km
Expanding the first part of the expression removes the first set of parentheses.
2
Distribute the negative sign across the trinomial (2k2−5km+m2)(2k^2 - 5km + m^2).
−2k2+5km−m2-2k^2 + 5km - m^2
Subtracting a polynomial requires reversing the sign of every term inside the parentheses.
3
Group and combine all like terms.
(4k2−2k2)+(−12km+5km)−m2=2k2−7km−m2(4k^2 - 2k^2) + (-12km + 5km) - m^2 = 2k^2 - 7km - m^2
Expressions are fully simplified when all terms with identical variable combinations are merged.

Key Concept

Distributing variables and negative signs followed by combining like terms.
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