The expression is equivalent to the polynomial , where , , , and are constants. What is the value of ?

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SAT Equivalent Expressions Question 224 | Free SAT Prep | Aniko

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Question 224·Hard·Equivalent Expressions

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The expression $(2x^2 - 7x + 5)(x + 4) - (x^2 - 3x - 2)(3x - 1)$ is equivalent to the polynomial $kx^3 + mx^2 + nx + p$ , where $k$ , $m$ , $n$ , and $p$ are constants.

What is the value of $n$ ?

Your answer

(Express the answer as an integer)

For expressions like this on the SAT, the fastest reliable method is to expand systematically and combine like terms while being very careful with signs, especially when one whole polynomial is subtracted from another. To save time when you only need a specific coefficient (like $n$ , the coefficient of $x$ ), focus only on the terms that can produce $x$ when you multiply: for each product, identify which pairs of terms give an $x$ -power of 1, find their combined linear coefficient, then account for the subtraction between the two products. This targeted approach reduces arithmetic and lowers the chance of mistakes with higher-degree or constant terms.

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Look at the structure of the expression

Notice the expression is one product minus another: $(2x^2 - 7x + 5)(x + 4)$ minus $(x^2 - 3x - 2)(3x - 1)$ . Think about how you usually simplify products of binomials and trinomials.

Expand each product separately

First, expand $(2x^2 - 7x + 5)(x + 4)$ into a single polynomial. Then, expand $(x^2 - 3x - 2)(3x - 1)$ separately. Do not combine them yet.

Be careful with the subtraction

After you have both polynomials, remember that the second one is being subtracted. That means every term in the second polynomial will change sign when you remove the parentheses.

Focus on the x-term

You only need the coefficient of $x$ . After expanding and distributing the minus sign, identify the terms that have $x$ (power 1) and combine just those coefficients to find $n$ .

Desmos Guide

Use Desmos to expand the expression

In an expression line, type:

expand((2x^2 - 7x + 5)(x + 4) - (x^2 - 3x - 2)(3x - 1))

Desmos will display the expanded polynomial in standard form.

Read off the coefficient of x

Look at the expanded polynomial that Desmos shows. Identify the term with $x$ (power 1), and note the number multiplying $x$ ; that number is the value of $n$ .

Step-by-step Explanation

Expand the first product

Start with $(2x^2 - 7x + 5)(x + 4)$ .

Distribute $x$ and $4$ across $2x^2 - 7x + 5$ :

Multiply by $x$ : $2x^2 \cdot x = 2x^3$ , $-7x \cdot x = -7x^2$ , $5 \cdot x = 5x$
Multiply by $4$ : $2x^2 \cdot 4 = 8x^2$ , $-7x \cdot 4 = -28x$ , $5 \cdot 4 = 20$

Add these results:

2x^3 - 7x^2 + 5x + 8x^2 - 28x + 20 = 2x^3 + x^2 - 23x + 20

So the first product simplifies to $2x^3 + x^2 - 23x + 20$ .

Expand the second product

Now expand $(x^2 - 3x - 2)(3x - 1)$ .

Distribute $3x$ and $-1$ across $x^2 - 3x - 2$ :

Multiply by $3x$ : $x^2 \cdot 3x = 3x^3$ , $-3x \cdot 3x = -9x^2$ , $-2 \cdot 3x = -6x$
Multiply by $-1$ : $x^2 \cdot (-1) = -x^2$ , $-3x \cdot (-1) = 3x$ , $-2 \cdot (-1) = 2$

Add these results:

3x^3 - 9x^2 - 6x - x^2 + 3x + 2 = 3x^3 - 10x^2 - 3x + 2

So the second product simplifies to $3x^3 - 10x^2 - 3x + 2$ .

Subtract the second polynomial from the first

The original expression is

(2x^2 - 7x + 5)(x + 4) - (x^2 - 3x - 2)(3x - 1).

Using the expanded forms from Steps 1 and 2, this becomes

(2x^3 + x^2 - 23x + 20) - (3x^3 - 10x^2 - 3x + 2).

Distribute the minus sign across the second parentheses:

2x^3 + x^2 - 23x + 20 - 3x^3 + 10x^2 + 3x - 2.

Now group like terms:

Cubic terms: $2x^3 - 3x^3$
Quadratic terms: $x^2 + 10x^2$
Linear terms: $-23x + 3x$
Constant terms: $20 - 2$

So the expression can be written as

(2x^3 - 3x^3) + (x^2 + 10x^2) + (-23x + 3x) + (20 - 2).

Combine like terms and identify n

Now simplify each grouped part:

Cubic: $2x^3 - 3x^3 = -x^3$
Quadratic: $x^2 + 10x^2 = 11x^2$
Linear: $-23x + 3x = -20x$
Constant: $20 - 2 = 18$

So the entire expression simplifies to

-x^3 + 11x^2 - 20x + 18.

This matches the form $kx^3 + mx^2 + nx + p$ , so $n$ is the coefficient of $x$ , which is $-20$ .

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Step-by-step Explanation

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Step-by-step Explanation