The polynomial has as a zero and is divisible by . The constants and are real. What is the value of ?

Solution available with step-by-step explanation

SAT Equivalent Expressions Question 102 | Free SAT Prep | Aniko

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Question 102·Hard·Equivalent Expressions

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The polynomial $4x^3 - 2p x^2 + qx - 18$ has $x = 3$ as a zero and is divisible by $2x + 1$ . The constants $p$ and $q$ are real. What is the value of $p + q$ ?

For polynomial questions with unknown coefficients where you are told certain zeros or factors, use the factor theorem: every given zero or factor gives you an equation by substituting that $x$ -value into the polynomial and setting it equal to 0. Aim to set up the fewest, simplest equations that directly involve the quantity asked for (such as $p+q$ ), rather than solving for every coefficient individually. Avoid long polynomial division; substitution and careful simplification are usually much faster and less error-prone on the SAT.

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Interpret the conditions correctly

Remember: if $x=a$ is a zero of a polynomial, then plugging $x=a$ into the polynomial gives 0. Likewise, if a polynomial is divisible by a linear factor like $2x+1$ , then the root of that factor is also a zero of the polynomial.

Find the zero that comes from $2x+1$

Solve $2x+1=0$ to find the $x$ -value associated with that factor. Use that $x$ -value in the polynomial and set the result equal to 0.

Look for an equation that combines $p$ and $q$

After substituting the correct $x$ -value, simplify carefully. Notice that the resulting equation involves both $p$ and $q$ together; rearrange it to isolate the combination $p+q$ directly, instead of solving for $p$ and $q$ separately.

Desmos Guide

Use Desmos to handle the substitution arithmetic

In one expression line, type:

4(-1/2)^3 - 2p(-1/2)^2 + q(-1/2) - 18

Desmos will simplify the numeric parts and show something like $-0.5p - 0.5q - 18.5$ . This corresponds to the symbolic equation $-\tfrac12 p - \tfrac12 q - \tfrac{37}{2} = 0$ when you set the expression equal to 0.

Use the simplified expression to isolate $p+q$

From the simplified Desmos expression, rewrite it as an equation equal to 0 and then, on paper, multiply through by 2 to clear fractions and rearrange to isolate $p+q$ . The value you get for $p+q$ is the one you should choose from the answer options; Desmos is helping you avoid arithmetic mistakes in the substitution step.

Step-by-step Explanation

Use what “divisible by $2x+1$ ” means

If a polynomial is divisible by a linear factor, then the root of that factor is a zero of the polynomial (this is the factor theorem).

The factor is $2x+1$ .
Solve $2x+1=0$ to find the corresponding zero: $x=-\tfrac12$ .

So $x=-\tfrac12$ is a zero, which means the polynomial must satisfy

4x^3 - 2p x^2 + qx - 18 = 0 \quad \text{when } x=-\tfrac12.

Substitute $x=-\tfrac12$ into the polynomial

Plug $x=-\tfrac12$ into the polynomial and set it equal to 0:

4\left(-\frac12\right)^3 - 2p\left(-\frac12\right)^2 + q\left(-\frac12\right) - 18 = 0.

Compute each term:

$\left(-\tfrac12\right)^3 = -\tfrac18$ , so $4\left(-\tfrac18\right) = -\tfrac12$ .
$\left(-\tfrac12\right)^2 = \tfrac14$ , so $-2p\left(\tfrac14\right) = -\tfrac12 p$ .
$q\left(-\tfrac12\right) = -\tfrac12 q$ .

So the equation becomes

-\frac12 - \frac12 p - \frac12 q - 18 = 0.

Simplify the equation and solve for $p+q$

Combine the constants on the left:

-\frac12 - 18 = -\frac12 - \frac{36}{2} = -\frac{37}{2}.

So the equation is

-\frac12 p - \frac12 q - \frac{37}{2} = 0.

Multiply every term by 2 to clear the denominators:

-p - q - 37 = 0.

Rearrange to isolate $p+q$ :

-(p+q) = 37 \quad \Rightarrow \quad p+q = -37.

Thus, $p+q=-37$ , which corresponds to choice D.

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Step-by-step Explanation

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Step-by-step Explanation