5.2 - Polynomials, Linear Factors, and Zeros

5.2 - Polynomials, Linear Factors, and Zeros

Objectives

• To find zeros of a polynomial function in factored form

• To write a polynomial in factored form

• To write a polynomial from its zeros

• To determine multiplicity of a factor of a polynomial

Math Response

If I asked you to write

$$y=\left(x-2\right)\left(x^2-3x-10\right)$$  in its factored form, how would you do that?

Type answer here

Correct Answer: y=\left(x-2\right)\left(x+2\right)\left(x-5\right)

Deg^°

Rad

You can factor the trinomial in this example:

Zeros of a Function

The point(s) at which the function intersects the x-axis
A polynomial function of degree n can have at most n real zeros
An even degree polynomial has an even number of zeros (or no zeros)
An odd degree polynomial has an odd number of zeros

Multiple Choice

Determine the zeros of the polynomial function

$$y=\left(x-2\right)\left(x^2-3x-10\right)$$  

• $$-2,\ -5,\ 2$$  

• $$-2,\ 5,\ 2$$  

• $$-2,\ -3,\ -10$$  

• $$2,\ 3,\ 10$$  

Math Response

Write the polynomial in its factored form:

$$\left(x-1\right)\left(x^2+8x+15\right)$$  

Type answer here

Correct Answer: \left(x-1\right)\left(x+3\right)\left(x+5\right)

Deg^°

Rad

Multiple Choice

Determine the zeros of the polynomial function:

$$\left(x-1\right)\left(x^2+8x+15\right)$$  

• $$x=1,\ x=-3,\ x=-5$$

• $$x=1,\ x=3,\ x=5$$

• $$x=-1,\ x=3,\ x=5$$

• $$x=1,\ x=8,\ x=15$$

What if a factor appears more than once?

Open Ended

Write the polynomial  $$x^3-6x^2+9x$$  in its factored form and state the multiplicity of any factors that appear more than once.

We can also work backwards, to write a polynomial from a set of zeros.

Write a polynomial that has 2, -3, and 4 as zeros

Poll

Select the image that most closely represents how well you feel you undersatnd polynomials, linear factors, and zeros.