In the polynomial function p(x) = x^3 - 5x^2 + 2x + k, the constant k is chosen such that … — SAT

In the polynomial function $p(x) = x^3 - 5x^2 + 2x + k$ , the constant $k$ is chosen such that $p(x)$ is divisible by $x - 4$ . What is the value of $k$ ?

Answer: 8

Answer

The value of

k

is 8.

According to the Factor Theorem, a polynomial

p(x)

is divisible by

x - c

if and only if

p(c) = 0

. Since

p(x)

is divisible by

x - 4

, we must have

p(4) = 0

. Substituting

4

for

x

in the equation for

p(x)

yields

4^3 - 5(4)^2 + 2(4) + k = 0

. Simplifying the numerical terms gives

64 - 80 + 8 + k = 0

, which simplifies further to

-8 + k = 0

. Therefore, the value of the constant

k

8

Step-by-Step Solution

Apply the Factor Theorem to relate the factor

x - 4

to the value of the polynomial.

Since

x - 4

is a factor of

p(x)

p(4) = 0

By the Factor Theorem, if

x - c

is a factor of a polynomial

p(x)

, then

p(c) = 0

Substitute

x = 4

into the polynomial expression.

4^3 - 5(4)^2 + 2(4) + k = 0

We substitute

x = 4

into

p(x) = x^3 - 5x^2 + 2x + k

and set the expression to

0

Simplify the numerical expression to solve for

k

64 - 80 + 8 + k = 0 \implies -8 + k = 0 \implies k = 8

Evaluate powers, multiply, and solve the resulting linear equation for

k

Key Concept

Factor Theorem

Estimated Time:1m 0s

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