Suppose you divide a polynomial by a binomial. How do you know if the binomial is a factor of the polynomial? Create a sample problem that has a binomial that is a factor of the polynomial being divided, and another problem that has a binomial that is not a factor of the polynomial being divided.

Answer 1

To determine if a binomial is a factor of a polynomial, you can use the factor theorem. The factor theorem states that a polynomial P(x) has (x - a) as a factor if and only if P(a) = 0.

Sample problem with binomial that is a factor:
Divide the polynomial 𝑥^3 - 8𝑥^2 + 19𝑥 - 12 by the binomial x - 2.

Since the binomial x - 2 is a factor, we can set x - 2 equal to 0 and solve for x:
x - 2 = 0
x = 2

Now plug x = 2 into the original polynomial:
P(2) = (2)^3 - 8(2)^2 + 19(2) - 12
P(2) = 8 - 32 + 38 - 12
P(2) = 2

Since P(2) = 0, the binomial x - 2 is a factor of the polynomial 𝑥^3 - 8𝑥^2 + 19𝑥 - 12.

Sample problem with binomial that is not a factor:
Divide the polynomial 2𝑥^3 - 6𝑥^2 + 4𝑥 - 8 by the binomial x + 2.

Since the binomial x + 2 is not a factor, we set x + 2 equal to 0 and solve for x:
x + 2 = 0
x = -2

Now plug x = -2 into the original polynomial:
P(-2) = 2(-2)^3 - 6(-2)^2 + 4(-2) - 8
P(-2) = 2(-8) - 6(4) - 8 - 8
P(-2) = -16 - 24 - 8 - 8
P(-2) = -56

Since P(-2) ≠ 0, the binomial x + 2 is not a factor of the polynomial 2𝑥^3 - 6𝑥^2 + 4𝑥 - 8.