In the expression above, is an integer. If the expression is equivalent to for all , which of the following could be the value of ?

Solution available with step-by-step explanation

SAT Equivalent Expressions Question 48 | Free SAT Prep | Aniko

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Question 48·Hard·Equivalent Expressions

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\frac{4x^3 + kx^2 - 9x - 27}{x + 3}

In the expression above, $k$ is an integer. If the expression is equivalent to $4x^2 - 9$ for all $x \ne -3$ , which of the following could be the value of $k$ ?

When a rational expression is said to be equivalent to a polynomial for all values of x (except where the denominator is zero), treat it as a polynomial identity. Multiply both sides by the denominator to eliminate the fraction, then either expand and match coefficients (the numbers in front of powers of x) or factor systematically to solve for unknown constants like k. This is usually faster and more reliable than plugging in many test values, and it avoids mistakes from trying to substitute values where the denominator is zero.

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Use the fact that the expressions are equal for all x

If a fraction equals a polynomial for all $x \ne -3$ , what must be true about the numerator and the denominator? Try multiplying both sides of the equation by $x+3$ .

Write the numerator as a product

After you multiply both sides by $x+3$ , you should get an equation where the numerator $4x^3 + kx^2 - 9x - 27$ equals $(x+3)$ times $4x^2 - 9$ . Focus on expanding this product correctly.

Match the $x^2$ terms

Once you expand $(x+3)(4x^2 - 9)$ , compare it to $4x^3 + kx^2 - 9x - 27$ . Look specifically at the coefficient (number in front) of $x^2$ to determine $k$ .

Desmos Guide

Enter the general expression with a slider for k

In Desmos, type k = 0 on a new line to create a slider for $k$ . On the next line, enter (4x^3 + kx^2 - 9x - 27)/(x + 3) to graph the rational expression.

Graph the target polynomial

On another line, type 4x^2 - 9 to graph the quadratic expression.

Test the answer choices using the slider

Move the $k$ slider to each of the answer choice values one at a time. For the correct value of $k$ , the graph of (4x^3 + kx^2 - 9x - 27)/(x + 3) will lie exactly on top of the graph of 4x^2 - 9 for all $x$ except at $x = -3$ , where the rational expression is undefined.

Step-by-step Explanation

Clear the denominator to relate the numerator to $4x^2 - 9$

We are told

\frac{4x^3 + kx^2 - 9x - 27}{x + 3} = 4x^2 - 9

for all $x \ne -3$ . Since $x+3 \ne 0$ for these $x$ , we can multiply both sides by $x+3$ :

4x^3 + kx^2 - 9x - 27 = (x+3)(4x^2 - 9).

Now the problem becomes: find $k$ so that this polynomial equality is true for all $x$ .

Expand the product on the right-hand side

Expand $(x+3)(4x^2 - 9)$ by distributing:

First distribute $x$ :

x(4x^2 - 9) = 4x^3 - 9x

Then distribute $3$ :

3(4x^2 - 9) = 12x^2 - 27

Add these results:

(x+3)(4x^2 - 9) = 4x^3 - 9x + 12x^2 - 27 = 4x^3 + 12x^2 - 9x - 27.

Match coefficients to find k (and the correct answer)

From Step 1, we need

4x^3 + kx^2 - 9x - 27 = 4x^3 + 12x^2 - 9x - 27.

For two polynomials to be equal for all $x$ , the coefficients (the numbers in front of each power of $x$ ) must match term by term. The $x^2$ term on the left is $kx^2$ , and on the right it is $12x^2$ , so we must have

k = 12.

Therefore, the correct answer is 12 (choice C).

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation