Question

T(x) = x^3 - 5x^2 - 9x + 45: x - 5

show that binomial is a factor of the polynomial. then factor the polynomial completely.​

Answer 1

Factors of the polynomial are: (x - 3)(x + 3)(x - 5).

What if factoring polynomial?

A polynomial with coefficients in a certain field or in integers is expressed as the product of irreducible factors with coefficients in the same domain by the process of factorization of polynomials, also known as polynomial factorization.

Given:

We have to show tat x - 5 is a factor of the polynomial and to find the factor of the polynomial.

First to show x - 5 is a factor of the given polynomial.

We know that,

If x - 5 is the factor of the given polynomial then by factor theorem x - 5 = 0.

So, x = 5

Plug x = 5 in t(x).

t(5) = 0

That means 5 is the zero of the polynomial t(x).

Therefore, x - 5 is the factor of the polynomial.

Now, to factor the polynomial.

Therefore, after factoring the polynomial we get, .

Answer 2

The steps on how to show that the binomial is a factor of the polynomial and then factor the polynomial completely are:

1. Set the binomial equal to 0.

x - 5 = 0

2. Solve for x.

x = 5

3. Substitute the value of x into the polynomial.

t(5) = 5^3 - 5(5^2) - 9(5) + 45

t(5) = 125 - 125 - 45 + 45

t(5) = 0

4. Since the value of the polynomial is 0 when x = 5, the binomial (x - 5) is a factor of the polynomial.

5. To factor the polynomial completely, we can use the difference of squares factorization.

t(x) = (x - 5)(x^2 + 9)

The complete factorization of the polynomial is:

t(x) = (x - 5)(x^2 + 9)