The polynomial satisfies the following conditions: 1. When is divided by , the quotient is and the remainder is the constant . 2. When is divided by , the remainder is . What is the value of ?

Solution available with step-by-step explanation

SAT Equivalent Expressions Question 126 | Free SAT Prep | Aniko

00:00

Question 126·Hard·Equivalent Expressions

0 / 233

The polynomial $P(x) = 2x^3 + bx^2 + cx + 8$ satisfies the following conditions:

When $P(x)$ is divided by $x^2 + 3$ , the quotient is $2x + k$ and the remainder is the constant $r$ .
When $P(x)$ is divided by $x + 1$ , the remainder is $5$ .

What is the value of $b + c + k + r$ ?

For problems where a polynomial is divided by another polynomial and the quotient and remainder are given, avoid doing long division; instead, immediately write the identity $P(x) = (\text{divisor})(\text{quotient}) + \text{remainder}$ , expand, and match coefficients of like powers of $x$ to get simple equations for the unknown constants. Then, use the Remainder Theorem for any linear divisors (like $x+1$ ) to get additional equations by evaluating $P$ at specific points. Solve this small system carefully, watching signs in the constant term, and finally compute any requested combination (like $b + c + k + r$ ), using a quick calculator or mental math check if needed.

Hints

Click me!

Translate the division information into an equation

Use the idea that "polynomial = (divisor)(quotient) + remainder." Write $P(x)$ in terms of $(x^2+3)$ , $(2x+k)$ , and $r$ .

Compare coefficients after expanding

Expand $(x^2+3)(2x+k)$ and match coefficients with $2x^3 + bx^2 + cx + 8$ . Which coefficients give you $c$ directly, and how does $b$ relate to $k$ ?

Use the Remainder Theorem for $x+1$

The remainder when dividing by $x+1$ equals $P(-1)$ . Plug $x=-1$ into $P(x)$ , use the $c$ value you already found, and set the result equal to 5 to solve for $b$ .

Finish by solving for all variables

Once you know $b$ (and therefore $k$ ), use your constant-term equation from the first step to find $r$ , then add $b + c + k + r$ .

Desmos Guide

Define constants and the polynomial

After you solve for the constants algebraically, define them in Desmos by typing (with your values) b = 5, c = 6, and k = 5. Then define the polynomial by typing P(x) = 2x^3 + b x^2 + c x + 8.

Check the remainder for division by $x+1$

In a new line, type P(-1) and confirm that the value shown is 5, matching the given remainder when dividing by $x+1$ . If it is not 5, your values for $b$ and $c$ need to be rechecked.

Verify the quotient and remainder for $x^2+3$

Create the expression corresponding to divisor times quotient by typing (x^2 + 3)(2x + k) (using your value of $k$ ). Then type P(x) - (x^2 + 3)(2x + k). The graph of this difference should be a horizontal line (a constant value), and that constant is your remainder $r$ .

Use Desmos to add the four numbers

In a final line, type b + c + k + r with your computed values. The displayed result is the number you should choose among the answer options.

Step-by-step Explanation

Use the division algorithm with $x^2+3$

When $P(x)$ is divided by $x^2+3$ with quotient $2x+k$ and remainder $r$ (a constant), the division algorithm says

P(x) = (x^2+3)(2x+k) + r.

Expand the product:

(x^2+3)(2x+k) = 2x^3 + kx^2 + 6x + 3k.

P(x) = 2x^3 + bx^2 + cx + 8 = 2x^3 + kx^2 + 6x + 3k + r.

Match coefficients of like powers of $x$ :

$x^3$ term: already matches ( $2$ on both sides).
$x^2$ term: $b = k$ .
$x$ term: $c = 6$ .
Constant term: $8 = 3k + r$ , so $r = 8 - 3k$ .

So far we know $c = 6$ , $b = k$ , and $r$ in terms of $k$ . We still need $b$ (and thus $k$ ) and then $r$ .

Apply the Remainder Theorem with $x+1$

The problem tells us that when $P(x)$ is divided by $x+1$ , the remainder is $5$ . By the Remainder Theorem, that means

P(-1) = 5.

Now substitute $x=-1$ into $P(x) = 2x^3 + bx^2 + cx + 8$ . From Step 1 we already know $c = 6$ :

P(-1) = 2(-1)^3 + b(-1)^2 + 6(-1) + 8.

Compute carefully:

$2(-1)^3 = -2$
$b(-1)^2 = b$
$6(-1) = -6$
Constant term is $+8$

P(-1) = -2 + b - 6 + 8.

Combine the constant terms:

-2 - 6 + 8 = 0,

P(-1) = b.

Since the remainder is $5$ , we have $b = P(-1) = 5$ . Therefore $b = 5$ , and from Step 1, $k = b = 5$ as well.

Find $r$ and compute the required sum

Use the constant-term equation from Step 1: $8 = 3k + r$ . We now know $k = 5$ , so

8 = 3(5) + r \Rightarrow 8 = 15 + r \Rightarrow r = 8 - 15 = -7.

We now have

$b = 5$
$c = 6$
$k = 5$
$r = -7$ .

Compute the sum:

b + c + k + r = 5 + 6 + 5 + (-7) = 5 + 6 + 5 - 7 = 9.

So the value of $b + c + k + r$ is 9.

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation