The polynomial function p is defined by p(x) = x^3 - 4x^2 - kx + 36, where k is a constant… — SAT

The polynomial function $p$ is defined by $p(x) = x^3 - 4x^2 - kx + 36$ , where $k$ is a constant. In the $xy$ -plane, the graph of $y = p(x)$ has an $x$ -intercept at $(3, 0)$ . What is the value of $k$ ?

Answer: 9

Answer

The correct answer is 9.

x

-intercept at

(3, 0)

indicates that when

x = 3

p(x) = 0

. Substituting

x = 3

and

p(3) = 0

into the equation

p(x) = x^3 - 4x^2 - kx + 36

results in

3^3 - 4(3)^2 - k(3) + 36 = 0

. Simplifying the terms gives

27 - 36 - 3k + 36 = 0

. The

-36

and

+36

terms cancel, leaving

27 - 3k = 0

. Solving for

k

gives

3k = 27

, which results in

k = 9

Step-by-Step Solution

Relate the

x

-intercept to the root of the polynomial function.

p(3) = 0

Since the graph of

y = p(x)

has an

x

-intercept at

(3, 0)

, the value of the function at

x = 3

must be

0

Substitute

x = 3

into the polynomial expression.

3^3 - 4(3)^2 - k(3) + 36 = 0

By setting the expression equal to

0

, we can solve for the unknown constant

k

Simplify the expression and solve for

k

k = 9

27 - 36 - 3k + 36 = 0

simplifies to

27 - 3k = 0

, which yields

3k = 27

and thus

k = 9

Key Concept

Relationship between x-intercepts of a graph and the roots of the polynomial function.

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