The equation above is true for all real numbers , where , , and are constants. What is the value of ?

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SAT Equivalent Expressions Question 150 | Free SAT Prep | Aniko

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Question 150·Hard·Equivalent Expressions

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(x^2 + 4x + c)(2x - d) + 3 = 2x^3 + 11x^2 + px + 7

The equation above is true for all real numbers $x$ , where $c$ , $d$ , and $p$ are constants. What is the value of $c + d + p$ ?

For polynomial identity questions where an expression equals another for all real $x$ , immediately think "match coefficients." First, expand and simplify the more complicated side into standard form, grouping like powers of $x$ . Then equate the coefficients of each power of $x$ to set up simple linear equations in the unknown constants. Solve these equations step by step (often starting with the constant and highest or next-highest power), and finally compute whatever combination of the constants the question asks for. This avoids plugging in many values of $x$ and is usually the fastest, most reliable method on the SAT.

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Think about what "true for all real x" implies

If two polynomial expressions in $x$ are equal for every real value of $x$ , what must be true about the coefficients of $x^3$ , $x^2$ , $x$ , and the constant terms on both sides?

Rewrite the left-hand side in standard polynomial form

Expand $(x^2 + 4x + c)(2x - d)$ carefully and then add the $+3$ . Collect like terms so that the left-hand side looks like $2x^3 + Ax^2 + Bx + C$ , where $A$ , $B$ , and $C$ involve $c$ and $d$ .

Set up equations for c, d, and p

Match your expressions for the $x^2$ , $x$ , and constant coefficients on the left-hand side with the numbers $11$ , $p$ , and $7$ on the right-hand side. Use the $x^2$ and constant equations first to find $d$ and $c$ , then use the $x$ coefficient equation to find $p$ , and finally add $c + d + p$ .

Desmos Guide

Verify your values of c, d, and p

In Desmos, enter your found values for $c$ , $d$ , and $p$ into the expressions:

Type y = (x^2 + 4x + c)(2x - d) + 3 with your numerical $c$ and $d$ .
On a new line, type y = 2x^3 + 11x^2 + p*x + 7 with your numerical $p$ . If both graphs lie exactly on top of each other for all visible $x$ -values, your values of $c$ , $d$ , and $p$ are consistent with the equation.

Compute c + d + p

In another Desmos line, type your specific numbers in the expression c + d + p (for example, something like 4/3 - 3 + 44/3, using the values you solved for). The single number that Desmos outputs is the value of $c + d + p$ .

Step-by-step Explanation

Use the idea of matching coefficients

The equation is true for all real numbers $x$ , which means the two polynomials are identical.

For two polynomials to be identical, the coefficients of each power of $x$ (the $x^3$ , $x^2$ , $x$ , and constant terms) must be equal on both sides. We will:

Expand the left-hand side into standard form
Match coefficients with the right-hand side
Solve for $c$ , $d$ , and $p$ and then add them.

Expand the left-hand side

First expand $(x^2 + 4x + c)(2x - d)$ .

\begin{aligned} (x^2 + 4x + c)(2x - d) &= x^2(2x - d) + 4x(2x - d) + c(2x - d) \\ &= 2x^3 - dx^2 + 8x^2 - 4dx + 2cx - cd \\ &= 2x^3 + (8 - d)x^2 + (2c - 4d)x - cd. \end{aligned}

Now include the $+3$ at the end of the original left-hand side:

(x^2 + 4x + c)(2x - d) + 3 = 2x^3 + (8 - d)x^2 + (2c - 4d)x - cd + 3.

So the left-hand side in standard form is

2x^3 + (8 - d)x^2 + (2c - 4d)x + (-cd + 3).

The right-hand side is already in standard form:

2x^3 + 11x^2 + px + 7.

Match coefficients to find d and c

Match coefficients of the same powers of $x$ from both sides:

Coefficient of $x^3$ : $2$ on both sides (this just checks out automatically).
Coefficient of $x^2$ : $8 - d = 11$ .
Coefficient of $x$ : $2c - 4d = p$ .
Constant term: $-cd + 3 = 7$ .

Use the $x^2$ and constant equations to find $d$ and $c$ :

From $8 - d = 11$ :

8 - d = 11 \Rightarrow -d = 3 \Rightarrow d = -3.

From $-cd + 3 = 7$ :

-cd + 3 = 7 \Rightarrow -cd = 4 \Rightarrow cd = -4.

Substitute $d = -3$ into $cd = -4$ :

(-3)c = -4 \Rightarrow -3c = -4 \Rightarrow c = \frac{4}{3}.

Now we know $c$ and $d$ ; next we will use them to find $p$ and then compute $c + d + p$ .

Find p and then compute c + d + p

Use the $x$ -coefficient equation $2c - 4d = p$ with $c = \frac{4}{3}$ and $d = -3$ :

\begin{aligned} p &= 2c - 4d \\ &= 2\cdot\frac{4}{3} - 4(-3) \\ &= \frac{8}{3} + 12 \\ &= \frac{8}{3} + \frac{36}{3} \\ &= \frac{44}{3}. \end{aligned}

Now compute $c + d + p$ :

\begin{aligned} c + d + p &= \frac{4}{3} + (-3) + \frac{44}{3} \\ &= \left(\frac{4}{3} + \frac{44}{3}\right) - 3 \\ &= \frac{48}{3} - 3 \\ &= 16 - 3 \\ &= 13. \end{aligned}

So the value of $c + d + p$ is $13$ .

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Step-by-step Explanation

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Step-by-step Explanation