Roots of Polynomials

Roots of Polynomials

by jill parker

​Solving by factoring

​Here is an example on how to factor using the grouping technique.

Multiple Choice

Factor completely:
2x³ + 5x² - 8x - 20

• (x² - 4)(2x + 5)
• (2x - 5)(x² + 4)
• (2x² - 4)(x + 5)
• (x - 2)(x + 2)(2x + 5)

Multiple Choice

WHAT FACTORING TECHNIQUE do YOU HAVE TO USE TO SOLVE FOR THE FACTORS OF 27a³-729?

• FACTORING BY GCF

• DIFFERENCE OF TWO SQUARES

• DIFFERENCE OF TWO CUBES

• DIAMOND AND BOX METHOD

Multiple Choice

What are the remaining factors for

(x³ +x² -17x +15) if (x -1) is a factor? (hint: use synthetic division)

• (x-1) (x+5) (x-3)

• (x-1) (x+5) (x + 3)

• (x+1) (x+5) (x-3)

• (x+1) (x-5) (x-3)

Multiple Choice

What is the first step in solving 
(x + 2)(x - 3) = 0      ?

• Plug in zero for x
• Solve for x
• FOIL
• Set each binomial factor equal to zero

​Solve by Factoring

​Here is an example on how to solve a polynomial by factoring

Multiple Choice

What are the solutions to this equation? (x-3)(x+2)=0

• x=3,-2
• x=-3,2
• x=2,3
• x=3,2

Multiple Choice

What are the zeros in (x-5)(x+4)

• 4 and -5

• 5 and -4

• -4 and -5

• no zeros

Multiple Choice

Write x³ + 9x² + 18x in factored form

• x(x + 6)(x + 3)

• x(x + 9)(x + 2)

• x(x + 6)

• x(x - 6)(x - 3)

Multiple Choice

Solve x² + x – 72 = 0.

• x = -8

• x = -8 or x = 9

• x = 8 or x = -9

• x = 9

Multiple Choice

Solve b²+25b=0

• b= 0 and b = 25

• b= 0 and b= -25

• b= -5 and b= 5

• b=5 an b=-2.5

​Remember synthetic division?

​The example shows a reminder of how to use synthetic division. You may have to refer to your notes for further examples.

Multiple Choice

Is (x-1) a factor of f(x)= x³-8x²+14x-4?

• Yes, (x-1) is a factor. There is a remainder.

• No, (x-1) is not a factor. The remainder is zero.

• Yes, (x-1) is a factor. The remainder is zero.

• No, (x-1) is not a factor. There is a remainder.

Multiple Choice

What are the three factors for P(x) = (x³ +x² -17x +15)?

• (x-1) (x+5) (x-3)

• (x-1) (x+5) (x + 3)

• (x+1) (x+5) (x-3)

• (x+1) (x-5) (x-3)

Multiple Choice

Decide if (x+3) is a factor of 3x³+10x²-x-12

• Yes, it is a factor!

• No, it is not a factor!

Multiple Choice

Solve for the values of x:

(2x - 1)(x + 4) = 0

• x = 1/2, -4

• x = 1/2, 4

• x = 2, -4

• x = -2 4

Multiple Choice

Find the correct statement about the zeros.

• −5 is a zero with multiplicity 1

• −5 is a zero with multiplicity 2

• −2 is a zero with multiplicity 1

• 2 is a zero with multiplicity 2

Multiple Choice

What are the roots of the polynomial function?
y = (x + 2)(x - 7)³

• -2, 7 multiplicity 3
• -2, 7, 3
• 2, -7 multiplicity 3
• 2, -7, 3

Multiple Choice

Which of the following is a COMPLETE  list of all possible Rational Zeros?
f(x) = x³ + 2x² - 6x + 8

• ±1, ±8
• ±1, ±2, ±4, ±8
• ±1, ±2, ±4
• 1, 2, 4, 8

Multiple Choice

How many zeros does the following function have?
f(x)= x⁵ - 3x³ + x

• 8
• 9
• 5
• 3

Multiple Choice

• (-2.55, 0), (2.55, 0)
• (-3, 0), (-2, 0), (2, 0), (3, 0)
• (0, 36)
• (0, -3), (0, -2), (0, 2), (0, 3)

Multiple Choice

What are the
zeros of the polynomial
y = 3(x + 4)(x + 1)(x - 3)?

• -4, -1, 3
• 3, -3, 4, 1
• 4, 1, -3
• 3, -4, -1, 3

Multiple Choice

Identify the zeros:

• x=-2, x=-1, x=0
• x=-2, x=0
• x=-1, x=0, x=2
• x=-2, x=0, x=1

Multiple Choice

Which of the following could be the equation of this graph in factored form? (Careful-pay attention to multiplicity.)

• f(x) = (x-4)(x-1)²(x+2)(x+4)
• f(x) = (x-4)(x+1)²(x+2)(x+4)
• f(x) = (x-4)(x-1)²(x-2)(x+4)
• f(x) = (x+4)(x+1)²(x-2)(x-4)

Multiple Select

Check any statements below that are true about the graph provided.

• This polynomial is degree 3

• X= 0 is a root

• The factor (x-2)^2 would be part of the equation

• X = 3 is a root

• This polynomial has a negative lead coefficient.

Multiple Choice

Factor: x⁴-x³-9x²+9x=0 when -3 is a root

• x(x-3)(x-1)(x+1)=0
• 2x(x-3)(x+2)(x+3)=0
• x(x-3)(x-1)(x+3)=0
• x(x-3)(x+1)(x+3)=0

Poll

How good do you feel about finding zeros (solutions) to a polynomial equation?

• I know it cold!

• I am almost there.

• Somewhat happy about it, maybe watch another video.

• I need to review my notes, watch a video and do more practice.

​On the last slide is a web page which shows a unique way to find zeros of a polynomial. Probably not the way it is done in your notes, but worth a look if you understand the process.

Finding Zeros of a Polynomial - High School Math

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