Directions: Use the distributive property to rewrite each expression as a binomial. Don't forget to draw your arrows and show any work.

$5(2x + 4) =$
$8(5x + 6) =$
$-3(5 - x) =$
$-(2x - 5) =$
$4x(x + 5) =$
$-2(9 - 2x) =$
$\frac{36x + 21}{3} =$
$\frac{3(4x + 10)}{6} =$

Directions: Simplify, show all steps.

$7(2x + 1) - 2(3x + 4)$
$2(x + 3) + 4(x - 1) + 10$

Question

03-11-2026
Mathematics

Answered

Directions: Use the distributive property to rewrite each expression as a binomial. Don't forget to draw your arrows and show any work.

$5(2x + 4) =$
$8(5x + 6) =$
$-3(5 - x) =$
$-(2x - 5) =$
$4x(x + 5) =$
$-2(9 - 2x) =$
$\frac{36x + 21}{3} =$
$\frac{3(4x + 10)}{6} =$

Directions: Simplify, show all steps.

$7(2x + 1) - 2(3x + 4)$
$2(x + 3) + 4(x - 1) + 10$

Verified Answer :

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-03T15:50:43-05:00

Apply the distributive property $a(b+c) = ab + ac$ to clear parentheses in all expressions.
Divide each term of a fraction numerator by the denominator to simplify rational expressions.
Combine like terms by adding or subtracting their coefficients after expansion.
The final simplified forms are: 4. $-2x+5$, 5. $4x^2+20x$, 6. $4x-18$, 7. $12x+7$, 8. $2x+5$, 9. $8x-1$, 10. $6x+12$.

Explanation

Understanding the Distributive Property and Approach

We are tasked with simplifying various algebraic expressions using the distributive property. The distributive property states that for any real numbers $a, b, c$, we have $a(b + c) = ab + ac$. Additionally, when dealing with division, we can distribute the divisor: $\frac{ax + b}{c} = \frac{ax}{c} + \frac{b}{c}$. For expressions involving multiple terms, we distribute first, then combine like terms to simplify.

Simplifying Expressions 4 through 8

To simplify each expression, we apply the distributive property systematically:

For $-(2x - 5)$, we distribute the $-1$: $-(2x - 5) = -2x + 5$.
For $4x(x + 5)$, we distribute $4x$: $4x(x + 5) = 4x^2 + 20x$.
For $-2(9 - 2x)$, we distribute the $-2$: $-2(9 - 2x) = -18 + 4x$, which is equivalent to $4x - 18$.
For $\frac{36x + 21}{3}$, we divide each term by 3: $\frac{36x}{3} + \frac{21}{3} = 12x + 7$.
For $\frac{3(4x + 10)}{6}$, we first simplify the fraction $\frac{3}{6} = \frac{1}{2}$, or distribute the 3 first: $\frac{12x + 30}{6} = \frac{12x}{6} + \frac{30}{6} = 2x + 5$.

Simplifying Expressions 9 and 10

Now we simplify expressions 9 and 10 by distributing and then combining like terms:

For $7(2x + 1) - 2(3x + 4)$:

Distribute the terms: $7(2x) + 7(1) - 2(3x) - 2(4) = 14x + 7 - 6x - 8$.
Combine like terms ($14x - 6x$ and $7 - 8$): $8x - 1$.

For $2(x + 3) + 4(x - 1) + 10$:

Distribute the terms: $2x + 6 + 4x - 4 + 10$.
Combine like terms ($2x + 4x$ and $6 - 4 + 10$): $6x + 12$.

Conclusion and Summary of Results

We have successfully simplified all expressions:

$-(2x - 5) = -2x + 5$
$4x(x + 5) = 4x^2 + 20x$
$-2(9 - 2x) = 4x - 18$
$\frac{36x + 21}{3} = 12x + 7$
$\frac{3(4x + 10)}{6} = 2x + 5$
$7(2x + 1) - 2(3x + 4) = 8x - 1$
$2(x + 3) + 4(x - 1) + 10 = 6x + 12$

The results are consistent with the distributive property and algebraic simplification rules.

Examples

The distributive property is a fundamental tool for simplifying complex expressions in many practical fields. For example, in budgeting or project management, if you have several cost components like labor, materials, and overhead, and each is affected by a scaling factor or tax rate, the distributive property helps you aggregate these costs into a single expression. Imagine you have a contract that pays a base rate plus a bonus for three separate project phases; using distribution allows you to calculate the total payment simply and accurately without errors.