Polynomial Roots

I can find the roots of a polynomial

y = -x² + 4x - 3

Multiple Choice

How many roots does this quadratic equation have?

• 0

• 1

• 2

• unknown

The highest exponent in the polynomial is the maximum number of roots.

Multiple Choice

This polynomial must have an exponent that is at least to what power?

• 2

• 3

• 5

• 6

How can we find roots of a polynomial?

Make a table substituting in x=0, x=1, x=2, x=-1, x=-2 and maybe you'll be lucky

The sure way to find roots of a polynomial is to:

1 Factor the polynomial

2 Set each factor equal to zero and solve for the variable

Polynomial #1: Factor 4x² - 44x + 120

What number is in common with each above that we can factor out of each term?

We can factor out 4 to get

4x² -44x + 120 = 4(x² -11x + 30)

Factors of 30 and their sums are:

30,1 add to 31 // 15.2 add to 17

10,3 add to 13 // 5,6 add to 11 ****MATCH

the negative means are factors a

4(x-5)(x-6)

Multiple Choice

Polynomial #1 continued: What are my roots or solutions for x?

1 Factor the polynomial: 4(x-5)(x-6)

2 Set each factor to zero and solve for the variable

x-5=0 and x-6=0.

Solve for x by adding the needed number to both sides.

• -5, -6

• -5, 6

• 5, 6

• 5, -6

x - 5 = 0 , x - 6 = 0

x - 5 + 5 = 0+5 , x - 6 + 6 =0 + 6

so x =5, x=6 are my roots (or zeros)

Polynomial #1:

4x² -44x +30=

4(x-5)(x-6) will cross the x axis in 2 places at x=5 and x=6 so its graph might look like this

Polynomial #2:

Can we factor

2x² + 4x + 5?

(2x + 5)(x+1)

Multiple Choice

Can we factor this?

• Yes we can factor everything. There is always a solution

• No, there is no real solution

Fill in the Blanks

Polynomial #3: Please factor x² + 4x + 4 and fill in below:

(x ___)(x ___)

Multiple Choice

Polynomial #3:

x² + 4x + 4 = (x + 2)(x + 2)

We set it to zero: (x + 2)(x + 2)= 0. Then we set each parentheses (or factor) equal to zero. What are the roots?

• 0, 2

• 22

• -2

x + 2 = 0

x + 2 -2 = -2

x = -2

so x² + 4x + 4 has one root or zero = -2

Multiple Choice

Polynomial #4: Find the roots or zeros of x² + 9x + 18

Hint: Factor it. Then set each factor in parentheses = 0 and solve for x.

• 9, 18

• -2, -9

• -3, -6

• 9, 2

x² + 9x + 18

Factors of 18 and their sums

18, 1 = 19

9, 2 = 11

6, 3 = 9 so factors are (x + 6)(x + 3) = 0

so x + 6 -6 = 0-6 and x + 3-3 = 0-3

so x= -6................and x=-3

Polynomial #5: what are the roots of

6x³ + 6x² -36x = 6x(x - 2)(x + 3)?

Set each factor to zero and solve

6x = 0, x - 2 = 0, x + 3=0

Multiple Choice

Polynomial #5:

What are the roots of 6x³ + 6x² - 36 = 6x (x - 2)(x + 3)?

• 6, -2, 3

• 6, 6, -36

• 0, 2, -3

• 0, -2, 3

6x (x - 2) (x + 3) = 0

6x =0, x - 2=0, x + 3=0

x=0, x=2, x =-3

Multiple Choice

Polynomial #6: What are the roots of x² + 10x + 24?

Hint: Factor the polynomial and then set each factor equal to zero.

• -6, -4

• 5, 6

• -10, -12

• 4, 6

Polynomial #6:

x² + 10x + 24 = (x + 4)(x + 6)

x + 4=0, x + 6= 0

x=-4, x=-6 are the roots or zeros

Polynomial #7:

Let's go in reverse. If we had this graph, or you knew that the roots or zeros were 2, -4, write the equation

Multiple Choice

Polynomial #7:

If our roots are -4 and 2 then x=-4 and x=2.

Now add to both sides so that the number is on the same side of the equation. x+4=0 and x-2=0. What are the factors of this polynomial?

• (x + 4)(x + 2)

• (x - 4)(x + 2)

• (x - 4)(x -2)

• (x + 4)(x - 2)

Polynomial #7 summary:

A polynomial with roots -4, 2 has the factors (x + 4)(x - 2). Now multiply those binomials to put it in standard form.

x² - 2x + 4x - 8 = x² + 2x - 8

Multiple Choice

CHECK FOR UNDERSTANDING

a. The roots of a polynomial are where the graph of the polynomial crosses which axis?

• x axis

• y axis

• both

Multiple Choice

b. The roots or zeros of a polynomial is where what equals zero?

• x coordinate of the y intercept point (0,y)

• y intercept

• y coordinate of the x intercept point (x, 0)

• None of the above

Multiple Choice

c. Polynomial #8:

What is the polynomial with roots of 0, -3?

• x(x+3) = x² + 3x

• (x - 0)(x -3) = x² - 3x

• x - 3

If the roots are x=0 and x=-3,

x =0, x + 3 = -3+3=0

(x)(x+3)=x(x+3) are my factors

Multiplying them gives x² + 3x

Multiple Choice

d. Which is true?

• The zeroes can always be found by factoring

• The degree (or highest exponent) tells you how many solutions there are

• It is always possible write a polynomial from its roots

Your ixl for today is Algebra2 K8 - at least to 70%.

After you have answered 2 correctly, you may continue offline. A video of this lesson will be posted with the Nov IXLs, next to K8]