Unit 3 IGCSE Polynomial Review

Unit 3 Polynomial Review IGCSE

​In this unit we look at the graphs and equations with degrees of 3 and higher

Coeffients are numbers that are multiplied times a variable

Constants are numbers not multipled times a variable​

Variables are letters that represent unkown numbers​

Terms are numbers, variables, or numbers times variables​

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​Vocabulary

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Multiple Choice

What is the constant in

$$7y^4-5x^3+9y^2-8x+4$$  

• 7

• -5

• -8

• 4

Multiple Choice

How many terms are in

$$3x^2\ +4x\ -\ 5$$  

• 1

• 2

• 3

• None

Multiple Choice

What is the coefficient of the linear term in the polynomial below?
8 + 5x 

• 8
• 0
• 5
• -5

Multiple Select

Select all the coefficients:

f(x) = 5x⁴+3x²+1

• 1

• 2

• 3

• 4

• 5

​-Degree of a term vs the degree of polynomial

​To find the degree of a term

add up all the exponents in term​

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To find the degree of the polynomial, identify the term with largest degree.

​

In this case, the polynomial is a 6th degree polynomial​

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​The term with the biggest power is often said to have the "leading coefficient"

but may or may not be the first term. ​

Multiple Choice

Identify the degree of: the term

2b⁸c²

• 10
• 8
• 2
• 16

Multiple Choice

What is the degree of the polynomial:

4x³ - 5x² + 2x - 1

• 1
• 2
• 3
• 4

Multiple Choice

What is the leading coefficient of the polynomial?

6x² + 7x⁴ + x³ + x

• 1

• 2

• 6

• 7

Polynomias are often described by the number of terms they have and their degree.

Multiple Choice

Classify by number of terms:

5x² – 6x + 3

• Monomial

• Binomial

• Trinomial

• 4-Term Polynomial

Multiple Choice

Classify by number of terms:
3x³ – 6x

• Monomial
• Binomial
• Trinomial
• 4-Term Polynomial

Multiple Choice

What is the degree of this polynomial?

2x⁴ - 3x⁵ + x

• -3

• 2

• 4

• 5

Multiple Choice

The degree of a polynomial can be determined by the largest powered monomial of the polynomial.  
What is the degree of the polynomial:
f(x) = 3x² + 4x - 5

• Degree 2  (quadratic)
• Degree 3 (cubic)
• Degree 4 (quartic)
• Degree 5 (quintic)

End behavior for a graph is what the y values are doing as the x's get infintely large and infinitely small.

​

Visually, it is what the graph is doing as you look to the right side and the left side.

​

​This graph has an end behavior of

down up​

or it may also be expressed this way​

​f(x) →∞ as x →∞

​f(x) →-∞ as x →-∞

Polynomial Function​ End Behavior ​

​The sign of the leading coefficient tells us what is happening to the y values on the right side of the graph​

​

The degree tells us whether the other end behavior is the same or not​

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​Leading Coefficient and Degree

​

​Some text here about the topic of discussion.

Multiple Choice

The sign of the leading coefficient determines the

behavior of the right end of the graph.  

The degree determines whether the left end behavior is the same or different.

find the end behavior : f(x) = 2x³ - x + 5

• Left up    

  Right up

• Left up  

  Right down

• Left down

  Right up

• Left down  

  Right down

Multiple Choice

Determine the end behavior

f(x) = 4 + 3x - 6x⁷ + 3x⁴

• UP and UP
• DOWN and DOWN
• UP and DOWN
• DOWN and UP

Multiple Choice

Describe the end behavior.
f(x) = - 2x³+3x²-5x+3

• f(x)→−∞ as x →−∞
  f(x)→−∞ as x →∞
• f(x)→∞ as x →−∞
  f(x)→∞ as x →∞
• f(x)→∞ as x →−∞
  f(x)→−∞ as x →∞
• f(x)→−∞ as x →−∞
  f(x)→∞ as x →∞

Multiple Choice

Describe the end behavior.
h(x) = x⁴-3x³ +2x-4

• f(x)→−∞ as x →−∞
  f(x)→−∞ as x →∞
• f(x)→∞ as x →−∞
  f(x)→∞ as x →∞
• f(x)→∞ as x →−∞
  f(x)→−∞ as x →∞
• f(x)→−∞ as x →−∞
  f(x)→∞ as x →∞

Multiple Choice

$$f\left(x\right)\ =-3\ \left(x-3\right)^2\left(x+4\right)$$  

Hint: to determine the degree of a polynomial in factored form, add up all of the multiplicities (exponents of factors with a variable)

The end behavior for this polynomial is...

Multiple Choice

The sign of the leading coefficient determines the

behavior of the right end of the graph.  

The degree determines whether the left end behavior is the same or different.

Which is true for the graph?

• Leading Coefficient Positive
  Degree - Even
• Leading Coefficient Positive
  Degree - Odd
• Leading Coefficient Negative
  Degree - Even
• Leading Coefficient Negative
  Degree - Odd

Multiple Choice

Which is true for the graph?

• Leading Coefficient Positive
  Degree - Even
• Leading Coefficient Positive
  Degree - Odd
• Leading Coefficient Negative
  Degree - Even
• Leading Coefficient Negative
  Degree - Odd

​When looking at a polynomial equation in standard form:

​

you can find the maximum number of turning points by taking the degree minus 1

Multiple Choice

Fill in the blank:  The maximum number of turning points is ____ less than the degree of the polynomial.

• 0
• 1
• 2
• 3

Multiple Choice

What is the maximum number of turning for a cubic function?

• 0
• 1
• 2
• 3

Multiple Choice

What is the maximum number of turning points for the function with a 5th degree?

• 1
• 2
• 3
• 4

you can reverse the process to determine the minimum degree of the function.

This function has two turning points, so that means that the minimum degree for the function would be 3​

​

the nmber of turning points Plus 1 will give you the minimum degree of the polynomial​

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​When given a graph...

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Multiple Choice

What is the degree of the function graphed here?

• 6
• 4
• 3
• 5

Multiple Choice

What is the minimum degree is the function?

• 1
• 2
• 4
• 10

To determine intervals of increasing, decreasing or constant:

-always look form left to right

-always give your interval with x values​

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​Intervals of increasing and decreasing

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To determine intervals of increasing, decreasing or constant:

-always look form left to right

-always give your interval with x values​

​

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​Intervals of increasing and decreasing

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Multiple Choice

Identify the increasing interval:

• (-∞, 1) U (3, ∞)
• (-∞, 4) U (3, ∞)
• (-∞, 1) U (0, ∞)
• (-5, 4) U (3, 5)

Multiple Choice

Identify the decreasing interval:

• (-∞, 1) U (3, ∞)
• (1, 3)
• (-∞, 1) U (0, ∞)
• (1,4) U (3,0)

The absolute or gobal maximum or minimum are the highest or lowest point on a graph.

​

A realtive (local) maximum is a high point in a small area​

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A realtive (local) minimum is a low point in a small area​​

​Local or Relative Maximum and Minimum

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Multiple Choice

Identify the relative maximum:

• (0, 3)
• (4, 1)
• (3, 0)
• (1, 4)

Multiple Choice

Identify the relative minimum:

• (0, 3)
• (4, 1)
• (3, 0)
• (1, 4)

The zeros of a function is the number or numbers that I can substitute in place of x that will cause the entire function to equal zero.

There are real and complex zeros.

The degree of a polynomial will give you the number of complex and real zeros​

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Complex zeros always come in pairs.​

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​Zeros of a function

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This graph has two turning points, so it has a minimum degree of 3.

​

This means there has to have at least 3 zeros.

​Real zeros can be found where the graph crosses the x axis.

The x intercept at -3 is a real zero.

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The other two zeros must be complex. ​

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​Real vs Complex Zeros

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Multiple Choice

How many real zeros does the function have?

• None
• 2
• 3
• 4

Multiple Select

Pick all statements that apply to this polynomial.

• It is an odd degree function.

• It is an even degree function.

• It has a positive leading coefficient.

• It has a negative leading coefficient.

• It has 3 real roots.

Multiple Choice

How many zeros do you expect the polynomial function
 P(x) = 3x³+4x-8 to have?

• 1
• 2
• 3
• 4

When a polynomial is written in factored form, the zeros are the number(s) that we could replace x with that would make any of the factors equal to zero​

​

The multiplicity of each zero, is how many times that factor happened. This can be found by looking at the exponent. ​

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​Zeros and Multiplicities

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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

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 zeroes 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

Write the polynomial in standard form given the following zeros. x = -2, 1, 4

• f(x) = (x+2)(x-1)(x-4)

• f(x) =x³ - 3x² - 6x + 8

• f(x) = x³ + 8

• f(x) = x³ - 3x² + 8

Multiple Choice

Write a 4th degree polynomial function with:
*a positive leading coefficient
*zeros: 9, 3, -2, 0

• y = x(x - 9)(x - 3)(x + 2)
• y = x(x + 9)(x + 3)(x - 2)
• y = -x(x - 9)(x - 3)(x + 2)
• y = -x(x + 9)(x + 3)(x - 2)

Real zeros and their multiplicities can be identified by looking where the graph intersects with the x axix

​

If the graph goes straight through the x axis, the multiplicity is one

if the graph bounces off the x axis the multiplicity is two​

If the graph flattens out and then goes straight through the x axis the multiplicity is three​

​Zeros and Multiplicities of polynomial graphs​

Multiple Choice

Which root has a multiplicity of 2?

• -1

• 2

• 4

• -4

Multiple Choice

What is the multiplicity of the zero 1.5

• 1
• 2
• 3
• 4

Multiple Choice

What are the zeros of the graph?

• -4 multiplicity 2, 0 , 2, 5

• -4 multiplicity 3, 0 , 2, 5

• 4 multiplicity 2, 0 , -2, -5

• 4 multiplicity 3, 0 , -2, -5

• 0, 11, -5 , 25

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)

​Polynomial Lond division

​

Typically used when the divisior is not just x plus a number.

​

Does Divide

McDonalds Multiply

Serve Subtract

Breakfast​ Bring down

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Then repeat until you run out of numbers​

Multiple Choice

$$\left(4x^4-8x^3-3x^2+7x-2\right)\div\left(2x-1\right)$$  

Use long division to evaluate

• $$2x^3-3x^2-3x+2$$  

• $$2x^3+3x^2+3x+2$$  

• $$2x^3-3x^2-3x-2$$  

• $$2x^3+2$$  

Multiple Choice

Identify the missing term to complete the solution.

• 4

• -4

• 5

• -5

Multiple Choice

• 4x - 2

• 4x + 2

• 4x - 2 + (1/3x - 1)

• 4x + 2 + (1/3x - 1)

​synthetic division

​

can be only be used when you are dividing a polynomial by

x + a number

or

x - a number​

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Multiply

Bring over

Add​

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​Don't forget to fill is a zero for any missing terms

Multiple Choice

Find the quotient

(2x³ − 5x² + 3x + 7) ÷ (x − 2)

• 2x³ − x² + x + 9

• 2x² − x + 1

• 2x² − x + 1 + 9/x−2

• 2x² − 9x − 15 − 23/x−2

Multiple Choice

What is the proper way to write the answer for the following problem?

• 2x³-x²-25x+12

• 2x⁴-x³-25x²-12x+0

• 2x³-x²-25x-12

• 2x⁴-x³-25x²-12x-0

Multiple Choice

Cam divided (x⁴ + 3x² - 4x - 2) by factor of (x-2) using synthetic division. His work is shown above. Which best describes his mistake?

• Cam wrote the remainder incorrectly.

• Cam did not use a zero place holder for the x³ term.

• Cam added instead of subtracting the rows.

• Cam should have used -2 as his division since the factor was x-2.

Multiple Choice

Which term is missing in this problem?

(2x³ + 5x² + 9) ÷

(x + 3)

• x⁴

• x³

• x²

• x

​Use this when you just need to know the remainder,

or

​whether a binomial is a factor

​because if the remainder is zero when you divide two polynomials, they must be factors.

​

Multiple Choice

What is the remainder when      a³ - 4 is divided by a+2? 

• -2
• -6
• -12
• 0

Multiple Choice

Is (x-3) a factor of
(x³ - 3x² + 2x + 2)?

• Yes
• No

Multiple Choice

What is the remainder when a³ - 4 is divided by a+2? 

• -2
• -6
• -12
• 0

Multiple Choice

            Is (x-2) a factor of             f(x)= x³-8x²+14x-4?

• Yes, (x-2) is a factor. There is a remainder.
• No, (x-2) is  not a factor. The remainder is zero.
• Yes, (x-2) is a factor. The remainder is zero.
• No, (x-2) is  not a factor. There is a remainder. 

Multiple Choice

when

f(x) = x⁴ − x³ + kx² − x + 1

is divided by x + 2, the remainder is 31. Find k

• −1

• 2

• 1

• −2

Multiple Choice

x + 1 is a factor of

f(x) = x³ + 2x² + k

Find k

• −1

• 2

• 1

• −2

​When adding polynomials:

​-combine like terms

​

​When subtracting polynomials:

​-distribute the negative 1 and then combine like terms

​

​When multiplying monomials:

​-multiply the coefficients and add the exponents.

Multiple Choice

f(x) = 3x² + 7x and g(x) = 2x² - x - 1,

find (f + g)(x).

• 11x² - 1
• 5x⁴ + 6x² - 1
• 5x² + 6x - 1
• 5x² + 8x - 1

Multiple Choice

Given  $$f\left(x\right)=3x^2-2x+1$$  and  $$g\left(x\right)=2x$$  ,

find  $$\left(f\times g\right)\left(x\right)$$  

• $$6x^2-4x-2$$  

• $$6x^3-4x^2+2x$$  

• $$6x^3+4x^2+2x$$  

• $$5x^3+4x^2-9x-4$$  

​Basic operations with Complex numbers

​adding and subtracting is just like adding and subtracting polynomials.

​

Multiplying is the same as multiplying polynomials except

i² = =1 ​

Multiple Choice

• 10
• -10
• 10i
• -10i

Multiple Choice

(3 + 2i) + (4 - 5i)

• 7 + 7i

• 1 - 3i

• 1 - 7i

• 7 - 3i

Multiple Choice

(4 - 2i) - (3 + 6i)

• 7 - 4i

• 1 + 4i

• 1 - 8i

• 7 - 8i

Multiple Choice

What does i² equal?

• -1

• √-1

• 1

• -√1

Multiple Choice

(2i)(3i)

• 5i
• -5
• 6i
• -6

Multiple Choice

The expression (2 + 3i)² is equal to

• -5
• -5 + 12i
• 13
• 13 + 12i

Multiple Choice

(3 + 8i)(-2 - i)

• 2-19i

• 23-i

• 34+5i

• 23-i