The polynomial function p is defined by p(x) = (x - 3)(x - 5)(x - r), where r is a constan… — SAT

The polynomial function $p$ is defined by $p(x) = (x - 3)(x - 5)(x - r)$ , where $r$ is a constant. If the graph of $y = p(x)$ in the $xy$ -plane intersects the $y$ -axis at $(0, -60)$ , what is the value of $r$ ?

Answer: 4

Answer

The value of

r

4

Evaluating the polynomial at

x = 0

yields the

y

-intercept of the graph, which is given as

-60

. Substituting

0

for

x

in the equation

p(x) = (x - 3)(x - 5)(x - r)

gives

p(0) = (0 - 3)(0 - 5)(0 - r) = (-3)(-5)(-r) = -15r

. Setting this expression equal to

-60

yields

-15r = -60

. Dividing both sides of the equation by

-15

yields

r = 4

Step-by-Step Solution

Identify the meaning of the

y

-intercept.

Evaluating

p(x)

x = 0

must equal

-60

, so

p(0) = -60

The

y

-intercept of a graph occurs where the

x

-coordinate is

0

Substitute

x = 0

into the polynomial expression.

p(0) = (0 - 3)(0 - 5)(0 - r) = (-3)(-5)(-r) = -15r

This simplifies the polynomial expression in terms of the unknown constant

r

Set the expression equal to the given

y

-intercept value and solve for

r

-15r = -60 \implies r = 4

Dividing both sides by

-15

isolates the variable

r

Key Concept

Using the y-intercept of a polynomial function to find an unknown root/constant in its factored form.

Question

Answer

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Key Concept

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