The cubic polynomial has the following properties: When is divided by , the remainder is . When is divided by , the remainder is . * When is divided by , the remainder is . What is the value of ?

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SAT Equivalent Expressions Question 63 | Free SAT Prep | Aniko

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Question 63·Hard·Equivalent Expressions

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The cubic polynomial $P(x) = x^3 + ax^2 + bx + c$ has the following properties:

When $P(x)$ is divided by $x - 1$ , the remainder is $3$ .
When $P(x)$ is divided by $x - 2$ , the remainder is $-5$ .
When $P(x)$ is divided by $x - 3$ , the remainder is $7$ .

What is the value of $a + b + c$ ?

For remainder problems with polynomials, immediately apply the Remainder Theorem: the remainder when dividing by $x-k$ is $P(k)$ . Write $P(k)$ by substituting that $x$ -value into the polynomial and set it equal to the given remainder. Then focus on the condition that most directly gives the quantity asked for (here, $P(1)$ gives $1+a+b+c$ ), instead of trying to solve for all coefficients. This saves time and reduces algebra errors.

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Use the Remainder Theorem

How is the remainder when dividing $P(x)$ by $x-k$ related to the value of the polynomial at $x=k$ ? Write each condition in terms of $P(1)$ , $P(2)$ , and $P(3)$ .

Focus on the simplest condition

You only need $a+b+c$ . Look at the condition involving $x-1$ . What is $P(1)$ in terms of $a$ , $b$ , and $c$ ?

Turn the condition into an equation

Once you write $P(1)$ as $1+a+b+c$ , set it equal to the given remainder. How can you isolate $a+b+c$ from that equation?

Desmos Guide

Use Desmos to compute the needed difference

From the equation $1 + a + b + c = 3$ , you need the value of $3 - 1$ . In Desmos, type 3 - 1 into the expression line and use the numerical output as the value of $a + b + c$ .

Step-by-step Explanation

Connect remainders to function values

Use the Remainder Theorem: when a polynomial $P(x)$ is divided by $x-k$ , the remainder is $P(k)$ .

So the three statements become:

$P(1)=3$
$P(2)=-5$
$P(3)=7$

We only need $a+b+c$ , so focus on the condition that involves $P(1)$ , because it will give a very simple expression.

Evaluate the polynomial at $x=1$

Substitute $x=1$ into $P(x)=x^3+ax^2+bx+c$ :

P(1) = 1^3 + a\cdot 1^2 + b\cdot 1 + c = 1 + a + b + c.

From the condition, the remainder when dividing by $x-1$ is $3$ , so $P(1)=3$ . This gives the equation

1 + a + b + c = 3.

Solve for the sum $a+b+c$

We want $a+b+c$ , and our equation is

1 + a + b + c = 3.

Subtract $1$ from both sides:

a + b + c = 3 - 1.

So $a + b + c = 2$ , which corresponds to choice C.

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Step-by-step Explanation

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Step-by-step Explanation