Roots and Zeros of Polynomial Functions

Roots of Polynomials

​There is useful information and examples on the slides preceding the questions. Each slide will literally TELL YOU how to do each problem and what to look for.

READ THE INFORMATION!!!

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READ THE INFO ON EACH SLIDE!!!

Roots of Polynomials

Names for roots

Polynomial functions have roots, which are also known as the x-intercepts, zeros, or solutions to the function. These are the "answers" when you factor and solve.

Multiple Choice

What are the different names to the solutions of a polynomial?

• x-intercepts

• roots

• zeros

• all of these

When a polynomial is in factored form, like on the right, we can just set each factor equal to 0 and solve the equation. In simple cases, we just switch the sign of the number in the parentheses.

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Solving a polynomial in factored form

Solve: (x+3)(x-2)(x+7)=0

Just switch the signs of the number in parentheses with x.

so... if x+3 is the factor, x=-3

... if x-2 is the factor, x=2

...if x+7 is the factor, x=-7

...if x is the factor, x=0

​Solve the factored functions on the following slides.

Multiple Choice

Solve: (x-8)(x+2)(x-5)=0

• x=-8

  x=2

  x=-5

• x=8

  x=2

  x=5

• x=8

  x=-2

  x=5

Fill in the Blank

Solve the following: (x-9)(x+7)(x+8)=0

Type your answers as x=1,2,-3.

(x=answers separated by commas with no spaces)

Fill in the Blank

Solve the following: x(x-2)(x+10)=0

Type your answers as x=1,2,-3.

(x=answers separated by commas with no spaces)

Zeros to Polynomials

We can take the given zeros of a function and write a polynomial in factored form.

Just do the opposite of what you did on the previous slides. We still switch the signs of our numbers, but now we put them with x inside a set of parentheses. For example:

If x=7, our factor is (x-7).

Multiple Choice

Select the correct factored form of a polynomial with zeros: -1, 2, and 5.

• $$\left(x-1\right)\left(x-2\right)\left(x-5\right)$$

• $$\left(x+1\right)\left(x+2\right)\left(x+5\right)$$

• $$\left(x-1\right)\left(x+2\right)\left(x+5\right)$$

• $$\left(x+1\right)\left(x-2\right)\left(x-5\right)$$

Fill in the Blank

Write a polynomial in factored form with the zeros:

-3,5,-7

(don't put spaces in your answer and keep the numbers in the same order given)

Fill in the Blank

Write a polynomial in factored form with the zeros:

2,5,-3

(don't put spaces in your answer and keep the numbers in the same order given)

Fill in the Blank

Write a polynomial in factored form with the zeros:

0,-8,5

(don't put spaces in your answer and keep the numbers in the same order given)

Number of Solutions

​The number of solutions to a function is given by its degree - the highest exponent.

Consider: f(x) = x²-4x+4... the highest exponent is 2, so there are 2 roots or 2 solutions.

If f(x)=x⁵+6x²-2x+1... the highest exponent is 5, so there are 5 roots.

Multiple Choice

Determine the number of solutions to the following polynomial.

$$f\left(x\right)=3x^2-10x+1$$

• 3

• 2

• 1

• 0

Multiple Choice

Determine the number of solutions to the following polynomial.

$$f\left(x\right)=x^7+5x^3-8x^2+9x-5$$

• 2

• 3

• 7

• 5

Fill in the Blank

Determine the number of solutions to the following polynomial:

$$f\left(x\right)=6x^8-4x^5+x-10$$

Fill in the Blank

Determine the number of solutions to the following polynomial:

$$f\left(x\right)=7x^2+3x^5-10+5x$$

Fill in the Blank

Determine the number of solutions to the following polynomial:

$$f\left(x\right)=3x-6$$

Sometimes, a polynomial has multiple roots at the same zero. When looking at a graph, there is an even multiplicity of the root if the curves bounces off the x-axis and there is an odd multiplicity the root if it crosses the x-axis.

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​Multiplicity

Multiple Choice

Each of these roots have an ______ multiplicity because the graph passes through the x-axis.

• even

• odd

Multiple Select

The polynomial has an even multiplicity at which root(s)? Select all that apply.

• x=-3

• x=2

• x=5

Multiple Select

The polynomial has an odd multiplicity at which root(s)? Select all that apply.

• x=-3

• x=2

• x=5

(x-2)⁴(x+3)²

The root 2 has a multiplicity of 4.

The root -3 has a multiplicity of 2.

​Examples

Multiplicity is shown in polynomial form as a degree (exponent). The exponent shows you what multiplicity a root has.

Multiplicity in Polynomials

Multiple Choice

In the polynomial $$f\left(x\right)=\left(x-2\right)\left(x+5\right)^2\left(x-3\right)^5$$ , the root x=-5 has a multiplicity of...

• 1

• 2

• 3

• 5

Multiple Choice

In the polynomial $$f\left(x\right)=\left(x-2\right)\left(x+5\right)^2\left(x-3\right)^5$$ , the root x=2 has a multiplicity of...

• 1

• 2

• 3

• 5

Multiple Choice

In the polynomial $$f\left(x\right)=\left(x-2\right)\left(x+5\right)^2\left(x-3\right)^5$$ , the root x=3 has a multiplicity of...

• 1

• 2

• 3

• 5

Fill in the Blank

In the polynomial $$f\left(x\right)=x^5\left(x+4\right)\left(x-5\right)^4$$ ,

the root x=5 has a multiplicity of...

Fill in the Blank

In the polynomial $$f\left(x\right)=x^5\left(x+4\right)\left(x-5\right)^4$$ ,

the root x=0 has a multiplicity of...

Fill in the Blank

In the polynomial $$f\left(x\right)=x^5\left(x+4\right)\left(x-5\right)^4$$ ,

the root x=-4 has a multiplicity of...