If (x + 3) is a factor of the polynomial P(x) = x^3 + 4x^2 + mx - 6, what is the value of … — JAMB UTME

If $(x + 3)$ is a factor of the polynomial $P(x) = x^3 + 4x^2 + mx - 6$ , what is the value of $m$ ?

Answer: 1

The value of

m

is 1.

According to the Factor Theorem,

(x + 3)

is a factor of

P(x)

P(-3) = 0

. Substituting

x = -3

into

P(x) = x^3 + 4x^2 + mx - 6

yields

(-3)^3 + 4(-3)^2 + m(-3) - 6 = 0

, which simplifies to

-27 + 36 - 3m - 6 = 0

, giving

3 - 3m = 0

and thus

m = 1

Apply the Factor Theorem

P(-3) = 0

By the Factor Theorem, for a linear divisor

(x - a)

to be a factor of

P(x)

P(a)

must equal zero. Here

x + 3 = 0 \implies x = -3

Substitute

x = -3

into

P(x) = x^3 + 4x^2 + mx - 6

(-3)^3 + 4(-3)^2 + m(-3) - 6 = 0

Evaluating

P(-3)

sets up an equation to find the unknown coefficient

m

Simplify and solve for

m

-27 + 36 - 3m - 6 = 0 \implies 3 - 3m = 0 \implies m = 1

Combine the constant terms

-27 + 36 - 6 = 3

and solve the linear equation in terms of

m

Factor Theorem

Question