A polynomial p is defined by p(x) = (x - 4)(x^2 + ax + 3), where a is a constant. If p(1) … — SAT

A polynomial $p$ is defined by $p(x) = (x - 4)(x^2 + ax + 3)$ , where $a$ is a constant. If $p(1) = -18$ , what is the value of $a$ ?

Answer: 2

Substituting

x = 1

into

p(x) = (x - 4)(x^2 + ax + 3)

gives

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

. Since we are given that

p(1) = -18

, we can set

-3(4 + a) = -18

. Dividing both sides of the equation by

-3

gives

4 + a = 6

. Subtracting

4

from both sides gives

a = 2

Substitute

x = 1

into the expression for

p(x)

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

We are given that

p(1) = -18

, so substituting

x = 1

will allow us to set up an equation in terms of the constant

a

Set the simplified expression equal to

-18

and solve for

a

a = 2

Dividing both sides of

-3(4 + a) = -18

-3

yields

4 + a = 6

, and subtracting

4

from both sides gives the value of

a

Using given function values to solve for unknown coefficients in a factored polynomial.

Question