If the polynomial p(x) = x^2 - kx + 12 is divisible by x - 3, where k is a constant, what … — SAT

If the polynomial $p(x) = x^2 - kx + 12$ is divisible by $x - 3$ , where $k$ is a constant, what is the value of $k$ ?

Answer: 7

According to the Factor Theorem, if a polynomial

p(x)

is divisible by

x - c

, then

p(c) = 0

. For the polynomial

p(x) = x^2 - kx + 12

to be divisible by

x - 3

, we must have

p(3) = 0

. Substituting

x = 3

into the polynomial gives

3^2 - k(3) + 12 = 0

. Simplifying this equation yields

9 - 3k + 12 = 0

, which combines to

21 - 3k = 0

. Solving for

k

gives

3k = 21

, which simplifies to

k = 7

Apply the Factor Theorem to find the root of the polynomial.

Since

x - 3

is a factor of

p(x)

, evaluating the polynomial at

x = 3

must equal 0, so

p(3) = 0

The Factor Theorem states that

x - c

is a factor of a polynomial

p(x)

if and only if

p(c) = 0

Substitute

x = 3

into

p(x) = x^2 - kx + 12

and set it equal to 0.

3^2 - k(3) + 12 = 0

This sets up an algebraic equation to solve for the unknown constant

k

Simplify the equation and solve for

k

9 - 3k + 12 = 0 \implies 21 - 3k = 0 \implies 3k = 21 \implies k = 7

Combine like terms and isolate

k

using basic algebraic operations.

The Factor Theorem states that a polynomial

p(x)

has a factor

(x - c)

if and only if

p(c) = 0

Question