Remainder and Rational Root Theorem

2.5 Factor and Remainder

Theorem

●Explain why when (x-k) is a factor of a polynomial, x=k is a zero of the
polynomial

●Interpret the remainder of a polynomial divided by (x-k)

●Given one factor of a polynomial function, use division to find the remaining
factors

Multiple Choice

What is (x² + 2x − 15) divided by (x + 5)?

• x+2

• x+3

• x-3

• $$\frac{x^2+2}{5x-3}$$

Multiple Choice

What is (2x⁵ - 5x⁴ + 7x³ +4x² - 10x + 11) divided by (x³ + 2)

Multiple Choice

What is the remainder of (2x³ - x² - 13x + 9) divided by (x - 2)

• -7

• -5

• 7

• 5

Let’s check it out:

f(-1) = 3(-1)3+(-1)2+2(-1)+5
f(-1) = -3 + 1 -2 + 5
f(-1) = 1

If there is a
remainder then (-1) is
not a zero of the
polynomial.

Multiple Choice

Try it now using the remainder theorem:

What is the remainder of (2x³ - x² - 13x + 9) divided by (x - 2)

• -7

• -5

• 7

• 5

Multiple Choice

If $$\left(9x^4-45x^3+37x^2+x+2\right)$$ is divided by $$\left(x-2\right)$$ then the remainder is...

• -64

• 64

• 656

• -652

Multiple Choice

If a polynomial p(x) is divided by x-a, then the remainder is obtained by:

• P(-a)

• P(a)

• P(a/2)

• P(-a/2)

Multiple Choice

If $$p\left(x\right)=x^3-ax^2+6x-a$$ is divided by $$\left(x-a\right)$$ , then the remainder is...

• 5

• -5

• a

• 5a

CYU