Super-Hard SAT Advanced Math Question 1 | Free SAT Prep

00:00

Try Full-Screen!

Question 1·25 Super-Hard SAT Advanced Math Questions·Advanced Math

0 / 25 Questions completed

The cubic function $p$ is defined by $p(x)=ax^3+bx^2+cx+d$ , where $a$ , $b$ , $c$ , and $d$ are constants and $a\ne0$ . The graph of $y=p(x)$ has $x$ -intercepts at $(-4,0)$ and $(2,0)$ , and $p(1)=p(-1)$ .

Which choice gives the value of $b$ in terms of $a$ ?

When a polynomial has known zeros, switch immediately from standard form to factored form. If another condition compares values of the function, substitute into that factored form before expanding. Once all factors are known, expand selectively or use coefficient patterns to obtain only the requested coefficient.

Hints

Click me!

Use the intercepts

An $x$ -intercept gives a zero of the polynomial. Write the cubic as a product containing $(x+4)$ and $(x-2)$ , along with one unknown linear factor.

Apply the equality carefully

Evaluate your factored expression at $x=1$ and at $x=-1$ . Set the resulting expressions equal to determine the unknown zero.

Avoid expanding everything

After finding the third factor, expand only enough to find the coefficient of $x^2$ . The $x^2$ terms come from choosing the constant from one factor and $x$ from the other two factors.

Desmos Guide

Use algebra first

Algebra is faster here because the factored form directly uses the two given zeros. Desmos can quickly verify the unknown third zero.

Graph the two evaluated expressions

Enter $y=-5(1-x)$ and $y=9(1+x)$ . Here, the graphing variable $x$ represents the unknown third zero $r$ after the common nonzero factor $a$ has been canceled.

Find the intersection

Click the intersection of the two lines. Its $x$ -coordinate gives the third zero. Then use $(x+4)(x-2)$ and the resulting third factor to check the $x^2$ coefficient.

Step-by-step Explanation

Write a factored form

Because $-4$ and $2$ are zeros, let the third zero be $r$ . Then

p(x)=a(x+4)(x-2)(x-r).

Use the given equality

Substitute $x=1$ and $x=-1$ into the factored form:

\begin{aligned} p(1)&=-5a(1-r) \\ p(-1)&=9a(1+r) \end{aligned}

Since these values are equal and $a\ne0$ ,

\begin{aligned} -5(1-r)&=9(1+r) \\ -5+5r&=9+9r \\ r&=-\frac{7}{2}. \end{aligned}

Identify the coefficient of $x^2$

The polynomial is now

p(x)=a(x+4)(x-2)\left(x+\frac{7}{2}\right).

First, $(x+4)(x-2)=x^2+2x-8$ . The $x^2$ coefficient after multiplying by $x+\frac{7}{2}$ is

2+\frac{7}{2}=\frac{11}{2}.

Therefore, $b=\frac{11}{2}a$ , which is choice C.

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation