A polynomial function p is defined by p(x) = ax(x-3)^2, where a is a constant. In the xy-p… — SAT

A polynomial function $p$ is defined by $p(x) = ax(x-3)^2$ , where $a$ is a constant. In the $xy$ -plane, the graph of $y = p(x)$ passes through the point $(1, -8)$ . What is the value of $p(2)$ ?

A
-25
B
-8
-4Answer
D
4

Answer

-4

The correct answer is -4. By substituting the point

(1, -8)

into the given function

p(x) = ax(x-3)^2

, we obtain

-8 = a(1)(1-3)^2

, which simplifies to

-8 = 4a

. Dividing both sides by

4

gives

a = -2

. Substituting

a = -2

back into the function yields

p(x) = -2x(x-3)^2

. Evaluating this function at

x = 2

gives

p(2) = -2(2)(2-3)^2 = -4(1) = -4

Step-by-Step Solution

Substitute the coordinates of the given point

(1, -8)

into the polynomial equation to set up an equation for the constant

a

p(1) = a(1)(1 - 3)^2 = -8

Since the graph passes through

(1, -8)

, substituting

x = 1

must yield

y = -8

Simplify the expression and solve for the constant

a

a(-2)^2 = -8 \Rightarrow 4a = -8 \Rightarrow a = -2

Evaluating

(1-3)^2

gives

4

, leading to a simple linear equation in terms of

a

Substitute

a = -2

back into the original polynomial definition to obtain the complete function formula.

p(x) = -2x(x - 3)^2

Knowing the value of the leading constant allows us to write the explicit formula for

p(x)

Evaluate

p(2)

by substituting

x = 2

into the completed function formula.

p(2) = -2(2)(2 - 3)^2 = -4(-1)^2 = -4

This calculation yields the value of the polynomial at

x = 2

as requested.

Key Concept

Polynomial Factors and Graphs

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