Question

Using substitution, determine whether the given linear expression is a factor of the polynomial. * 2x^3 - 11x^2 + 8x - 15; x - 5 YES OR NO

Answer 1

Given the ploynomial:

`\begin{gathered} 2x^3-11x^2+8x-15 \\ \\ \text{ Linear expression: x - 5} \end{gathered}`

A linear expression (x - n) is a factor of a polynomial when x = n is a zero of the polynomial expression. That is when the function f(x) = 0.

Given the linear expression:

x - 5

Equate to zero:

x - 5 = 0

Add 5 to both sides:

x - 5 + 5 = 0 - 5

x = 5

Substitute 5 for x in the polynomial, if the function tends to zero, then the linear expression is a factor of the polynomial.

We have:

`\begin{gathered} 2(5)^3-11(5)^2+8(5)-15^{} \\ \\ 2(125)-11(25)+8(5)-15 \\ \\ 250-275+40-15 \\ \\ -25+40-15 \\ \\ 15-15=0 \end{gathered}`

Thus:

`\begin{gathered} f(x)=2x^3-11x^2+8x-15 \\ \\ f(5)=2(5)^3-11(5)^2+8(5)-15^{} \\ \\ f(5)=0 \end{gathered}`

Since the the polynomial function f(x) tends to zero, the linear expression can be said to be a factor of the polynomial

ANSWER:

Yes, the linear expression is a factor of the polynomial