Polynomial Classifying by Degrees, Zeros, and Multiplicities

Polynomial Classifying by Degrees, Zeros, and Multiplicities

By Mrs. Bha

CLassifying a polynomial by degree

TAKE NOTES ON YOUR HANDOUT

Multiple Choice

Classify by degree of polynomial:
3x³ – 6x

• Constant

• Linear

• Quadratic

• Cubic

Multiple Choice

Classify by degree of polynomial:
12

• Constant

• Linear

• Quadratic

• Cubic

Multiple Choice

Classify by degree of polynomial:
7x³ – 8x² + 9

• Constant

• Linear

• Quadratic

• Cubic

Multiple Choice

Classify by degree of polynomial:
3x² – 8x + 1

• Constant

• Linear

• Quadratic

• Cubic

Multiple Choice

Classify by degree of polynomial: 7x³ – 8x⁴ -4x+ 9

• Quadratic

• Linear

• Quartic

• Cubic

THERE ARE MANY NAMES FOR THE Answers to a polynomial set equal to zero:

• solutions

• zeros

• roots​

• x-intercepts (where the graph passes the x-axis)​

Subject | Subject

Some text here about the topic of discussion

Find the zeros (X-INTERCEPTS AKA ROOTS)

y = (x+1)(x+2)​

0 = ​(x+1)(x+2)​ SET y EQUAL TO ZERO

TAKE NOTES ON YOUR HANDOUT

x+1 = 0

x= -1

x+2 = 0

x = -2

​The x-intercepts of the graph of this polynomial are: -1 and -2

Find the zeros (X-INTERCEPTS AKA ROOTS)

y = x(x-1)²(x+8)​

0 = ​x(x-1)²(x+8)​​ SET y EQUAL TO ZERO

TAKE NOTES ON YOUR HANDOUT

​

x= 0

​

x-1 = 0

x= 1

x+8 = 0

x = -8

​The x-intercepts of the graph of this polynomial are: 0, 1, -8

Find the zeros (X-INTERCEPTS AKA ROOTS)

​y = 5x³ +5x² -30x

0 = 5x(x² + x -6)​ = 0

0 = 5x(x+3)(x-2)​ = 0

​

TAKE NOTES ON YOUR HANDOUT

​5x = 0

x= 0​

x+3 = 0

x= -3

x-2 = 0

x = 2

​The x-intercepts of the graph of this polynomial are: 0, -3, and 2

Multiple Choice

What are the zeros of the function $$g\left(x\right)=\left(x+6\right)\left(x-4\right)\left(x+2\right)$$  ?

Hint: set each ( ) = 0 and solve

• $$-6,\ -4,\ -2$$  

• $$-4,\ 2,\ 6$$  

• $$-6\ -2,\ 4$$  

• $$2,\ 4,\ 6$$  

Multiple Choice

What are the zeros of the function $$\left(x-3\right)\left(x-7\right)\left(x+5\right)$$  ?

• $$-7,\ -5,\ -3$$  

• $$-7,\ -3,\ 5$$  

• $$-5,\ -3,\ 7$$  

• $$-5,\ 3,\ 7$$  

Multiple Choice

What are the zeros of the function $$f\left(x\right)=3x\left(x-4\right)\left(x+5\right)$$  ?

• $$-4,\ 0,\ 5$$  

• $$-4,\ 3,\ 5$$  

• $$-5,\ -3,\ 4$$  

• $$-5,\ 0,\ 4$$  

Multiple Select

What are the zeros for the given graph? Select all that apply.

• $$-1$$

• $$0$$

• $$1$$

• $$2$$

Multiple Select

What are the zeros of the given graph ? Choose all that apply.

• $$-3$$

• $$-1$$

• $$1$$

• $$3$$

Multiple Select

What are the zeros of the given graph? Select all that apply.

• $$3$$

• $$0$$

• $$1.5$$

• $$-3$$

Multiple Choice

Which one could be a representation of the equation of this graph? Is the leading coefficient positive or negative (look at the right arrow)

• $$-x\left(x-4\right)\left(x+3\right)\left(x+7\right)$$  

• $$-x\left(x+4\right)\left(x-3\right)\left(x-7\right)$$  

• $$x\left(x-4\right)\left(x+3\right)\left(x+7\right)$$  

• $$x\left(x+4\right)\left(x-3\right)\left(x-7\right)$$  

MULTIPLICITY

Graphs behave differently at various x-intercepts.

Sometimes the graph will cross over the x-axis at an intercept.

Other times the graph will touch the x-axis and bounce off.

​

TAKE NOTES ON YOUR HANDOUT

​Multiplicities are the

numbers 1, 2, 3, & 4 ->

and they are the exponents.

For 3, we also say "squiggles" thru x-axis

For 4, we also say "flat bounce"

MULTIPLICITY

TAKE NOTES ON YOUR HANDOUT

​Answer:​

Zero: x= -3 so (x+3)

Zero: x= -1 so (x+1)​

Zero: x= 4 so (x-4)​

​

​

y = (x+3)²(x+1)(x-4)³ multiplicity is the exponent​

​ bounce back, mult=2

​ crosses thru x-axis (linear), mult = 1

​ squiggles thru x-axis, mult=3

​​

​

​

MULTIPLICITY

TAKE NOTES ON YOUR HANDOUT

​

Answer:​

Zero: x= -5 so (x+5)squiggle thru x-axis, mult=3

Zero: x= -1 so (x+1)​ bounce back, mult=2

Zero: x= 3 so (x-3)​ flat bounce back, mult=4

​​

​

y = (x+5)³(x+1)²(x-3)​⁴

MULTIPLICITY

TAKE NOTES ON YOUR HANDOUT

​Which is the graph of h(x) = x²(x-3)?

​zeros are 0 and 3, so we can eliminate C.

• bounces at 0 (since multiplicity of x is 2)​

• crosses at 3 (​since multiplicity of x-3 is 1)

• Answer is B​

Multiple Choice

In the function $$m\left(x\right)=x^2\left(x+3\right)\left(x-3\right)\left(x+4\right)$$  which ZERO has a multiplicity of 2? Recall multiplicity is the exponent.

• $$-4$$  

• $$-3$$  

• $$0$$  

• $$3$$  

Multiple Choice

In the function $$p\left(x\right)=3x\left(x-6\right)^4\left(x+4\right)^3\left(x-5\right)^2$$  which zero has a multiplicity of 3?

• $$-4$$  

• $$0$$  

• $$5$$  

• $$6$$  

Multiple Select

What are the zeros of the given graph ? Choose all that apply.

• $$-3$$

• $$-1$$

• $$1$$

• $$3$$

Multiple Choice

What are the zeros?

• x = - 4

• There are none

• x = - 1.5 & 1.5

• x = - 3.5 and 3.5

Multiple Select

What are the zeros for the given graph? Select all that apply.

• $$-1$$

• $$0$$

• $$1$$

• $$2$$

Multiple Select

What are the zeros of the given graph? Select all that apply.

• $$-3$$

• $$0$$

• $$1.5$$

• $$3$$

Multiple Choice

Factor completely:
x³+7x²-2x-14

• (x²-2)(x-7)

• (x²+2)(x-7)

• (x²+2)(x+7)

• (x²-2)(x+7)

Multiple Choice

Factor Completely:

2n³ + 7n² - 2n - 7

• cannot be factored

• (n - 1)²(2n + 7)

• (n² - 1)(2n + 7)

• (n + 1)(n - 1)(2n + 7)

Multiple Choice

Factor completely: $$x^3-9x$$  

• $$x\left(x-3\right)$$  

• $$x\left(x+3\right)\left(x-3\right)$$  

• $$\left(x+3\right)\left(x-3\right)$$  

• $$x\left(x+3\right)$$  

Multiple Choice

Factor completely: $$2x^2-2x-12?$$  

• $$2\left(x+2\right)\left(x-3\right)$$  

• $$2\left(x+6\right)\left(x-1\right)$$  

• $$2\left(x+3\right)\left(x-2\right)$$  

• $$2\left(x+1\right)\left(x-6\right)$$  

Multiple Choice

Factor : 4p³+8p²+3p+6

• (2p²+3)(p+2)

• (2p²+6)(p+4)

• (4p²+3)(p+2)

• (4p²+8p)(3p+6)

Multiple Choice

Which one could be a representation of the equation of this graph?

• $$-x\left(x-4\right)\left(x+3\right)\left(x+7\right)$$  

• $$-x\left(x+4\right)\left(x-3\right)\left(x-7\right)$$  

• $$x\left(x-4\right)\left(x+3\right)\left(x+7\right)$$  

• $$x\left(x+4\right)\left(x-3\right)\left(x-7\right)$$  

Multiple Choice

• y = x(x - 3)(x - 2)

• y = x(x - 3)² (x+2)

• y = x(x + 3)² (x - 2)

• y = -x(x + 3)² (x - 2)

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 zeros with multiplicities?

$$g(x)=(x+4)^2(x+8)^3(x-5)$$  

• -4 multiplicity 1,

  -8 multiplicity 1,

  5 multiplicity 3

• 4 multiplicity 2,

  8 multiplicity 3,

  -5 multiplicity 1

• 4 multiplicity 3,

  8 multiplicity 2,

  -5 multiplicity 1

• -4 multiplicity 2,

  -8 multiplicity 3,

  5 multiplicity 1

Multiple Choice

What are the zeros with multiplicity?

$$k(x)=(x+2)^{10}(x-12)^9$$  

• -2 multiplicity 9,

  12 multiplicity 10

• -2 multiplicity 10, 12 multiplicity 9

• 2 multiplicity 10, -12 multiplicity 9

• 2 multiplicity 9, -12 multiplicity 10

Multiple Choice

What are the zeros with multiplicities?

$$h(x)=(x-2)(x-5)(x+6)$$  

• 2 multiplicity 1,

  5 multiplicity 1,

  6 multiplicity 1

• -2 multiplicity 1,

  -5 multiplicity 1,

  -6 multiplicity 1

• 2 multiplicity 1,

  5 multiplicity 1,

  -6 multiplicity 1

• -2 multiplicity 1,

  -5 multiplicity 1,

  6 multiplicity 1

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

Given  $$f\left(x\right)\ =\ \left(2x\ +6\right)^7\left(x-3\right)^4$$ , what is the multiplicity of the root  $$x=-3$$  ?

• 7

• 2

• 6

• 4

Multiple Choice

Select the polynomial whose zeros and degree are given.

Zeros: – 5(multiplicity of 3), 9(multiplicity of 2), – 2, 4.

• (x + 5)³(x - 9)²

  (x + 2)(x - 4)

• (x + 5)³(x - 9)²

  (x + 2)⁷(x - 4)⁷

• (x + 5)³(x - 9)²

  (x + 2)

• (x + 5)(x - 9)(x + 2)(x - 4)

Multiple Choice

Select the polynomial whose zeros and degree are given.

Zeros: – 5 (multiplicity of 3),

9 (multiplicity of 2),

– 2 (multiplicity of 1),

4 (multiplicity of 1)

• (x - 5)³(x + 9)²

  (x + 2)(x - 4)

• (x + 5)³(x - 9)²

  (x + 2)(x - 4)

• (x + 5)³(x - 9)²

  (x + 2)

• (x + 5)(x - 9)(x + 2)(x - 4)

Multiple Choice

In the above graph, which root has a multiplicity of 2?

• -1

• -3

• -120

• 4

Multiple Choice

How does a multiplicity of 2 affect the graph?

• crosses straight thru x-axis

• bounces off x-axis

• wiggle thru x-axiss