NC.M3.U2 Study Guide (Extra Practice)

​NC Math 3 Unit 2 Study Guide (Extra Practice)

​Use this Quizizz to study for our upcoming test on Monday, 9/30.

Fill in the Blank

What is the DEGREE of the polynomial

$$f\left(x\right)=3x^4+2x^3-7x^2-5x+9$$  

Fill in the Blank

What is the LEADING COEFFICIENT of the polynomial

$$f\left(x\right)=3x^4+2x^3-7x^2-5x+9$$  

Fill in the Blank

What is the DEGREE of the polynomial

$$g\left(x\right)=-4x^3-8x^2+2x-10$$  

Fill in the Blank

What is the LEADING COEFFICIENT of the polynomial

$$g\left(x\right)=-4x^3-8x^2+2x-10$$  

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Think Quadratics

​Concave up vs. Concave down

​Think Linear

​Positive slope vs. Negative slope

​Summary of End Behaviors

Multiple Choice

Does the graph show an EVEN or ODD degree polynomial?

• EVEN degree

• ODD degree

Multiple Choice

Does the graph have a POSITIVE or NEGATIVE leading coefficient?

• POSITIVE

• NEGATIVE

Multiple Choice

Does the graph show an EVEN or ODD degree polynomial?

• EVEN degree

• ODD degree

​Multiplicity

• A zero has MULTIPLICITY.

• ​Multiplicity is the number of times the zero's associated factor appears in the polynomials.

• ​You can determine if the graph will bounce or cross  the x-axis from the multiplicity  

  • ​If the multiplicity is odd, the graph will cross at that zero

  • ​If the multiplicity is even, the graph will bounce at that zero 

Multiple Select

Select all the zeros (with multiplicities) of

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

• x= -2 multiplicity 2

• x= 2 multiplicity 1

• x= -2 multiplicity 1

• x= -1 multiplicity 2

• x= 1 multiplicity 2

Multiple Select

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

From the last question, we know that f(x) has zeros at

x= -2 multiplicity 1 and x= 1 multiplicity 2.

Select whether the graph of f(x) will cross or bounce at each of the zeros.

• cross at x= -2

• bounce at x= -2

• cross at x=1

• bounce at x=1

Multiple Select

Select all the zeros (with multiplicities) of

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

• x= -4 multiplicity 2

• x= 4 multiplicity 2

• x= 0 multiplicity 2

• x= 2 multiplicity 2

• x= -2 multiplicity 2

They are also the values of x that, when plugged into a function, cause the value of the function to be zero.

​

Zeros are just x-intercepts

Multiple Choice

The graph of f(x) is shown. Which statement about f(x) is true?

• The zero of f(x) is -3

• The zero of f(x) is 2

• The zeros of f(x) are -1 and 1

• The zeros of f(x) are -6 and 6

The Zeros and Factors of a Function are related

The degree of the function matches the number of zeros that the function has.

Some zeros may be "nonreal" which means that they will not be visible on the x-axis.

The function below has 6 zeros because it is of degree 6.

Multiple Select

A polynomial function contains the factors x, x-3, and x+1. Which graphs could represent the polynomial function. SELECT ALL THAT APPLY

Multiple Choice

If the function $$p\left(x\right)$$ has a zero at 5, which of the following is a true statement?

• (x-5) is a factor of p(x)

• (x+5) is a factor of p(x)

• p(x) has a

  y-intercept at (0,5)

• p(x) has 3 real solutions

Multiple Choice

Which of the polynomials below *could* have at most 6 zeros?

• $$f\left(x\right)=4x^4-5x^2-14$$

• $$f\left(x\right)=-2x^6+4x^4-3x^2+2x-9$$

• $$f\left(x\right)=3x^5-4x^3+2x-9$$

Fill in the Blank

At most, how many zeros does the polynomial below have?

$$f\left(x\right)=5x^4-7x^2+1$$  

X-intercepts

Write an equation for the graph by writing the x-intercepts as factors

Multiple Choice

Which equation matches the graph?

• 
  

  $$y=\left(x+1\right)\left(x+2\right)$$
  

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

• $$y=x\left(x+2\right)$$

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

• $$y=x^2+3x+2$$

Multiple Choice

Which equation matches the graph?

• $$y=\left(x-3\right)\left(x+2\right)\left(x-1\right)$$

• $$y=\left(x-3\right)\left(x-2\right)\left(x-1\right)$$

• $$y=\left(x+3\right)\left(x-2\right)\left(x+1\right)$$

• $$y=x\left(x-1\right)\left(x+2\right)\left(x+3\right)$$

Math Response

Write the equation for the polynomial function that is graphed.

Type your answer as "y=" instead of "f(x)=".

Type answer here

Correct Answer: y=4\left(x-1\right)\left(x+1\right)\left(x+2\right)

Deg^°

Rad

Math Response

Write the equation for the polynomial function that is graphed.

Type your answer as "y=" instead of "f(x)=".

Type answer here

Correct Answer: y=\frac{1}{4}\left(x+1\right)^2\left(x+6\right)

Deg^°

Rad

Fill in the Blank

Find the value of c when the polynomial p(x) is divisible by (x + 2)

$$p\left(x\right)=4x^3+cx^2+x+2$$

Type just the number as your answer.

Fill in the Blank

Find the value for d in the polynomial f(x) if (x - 2) is a known factor of f(x).

$$f(x)=x^4-dx^3+24x^2-32x+16$$

Type your answer as just a number.

Multiple Choice

• 4x - 2

• 4x + 2

• 4x - 2 + (1/3x - 1)

• 4x + 2 + (1/3x - 1)

Multiple Choice

Is this division problem worked correctly?

• This is correct!
• This is incorrect!

Multiple Choice

What should you add to this problem?

(2x³ + 5x² + 9) ÷ (x + 3)

• 0x⁴

• 0x³

• 0x²

• 0x

Multiple Choice

Use long divison to solve

$$\left(6n^2-4n-16\right)\div\left(n-2\right)$$  

• n + 8

• 6n + 8

• 6n + 2

• n + 2

Multiple Choice

What is the final answer?

$$\frac{3x^3-4x^2-9x+10}{x-2}=?$$  

• $$3x^2+2x-5$$  

• $$3x^3+2x^2-5x+0$$  

​This video covers sums and differences of cubic expressions. On the next slide, I'll have the formulas that he shows on his whiteboard.

Sum or difference of perfect CUBES

Follow the pattern, and remember to use SOAP!

When you have an expression with only two terms, always look to see if there is a something you can factor out to make them perfect squares or perfect cubes.

Multiple Choice

Which of the following is the rule for factoring the sum of cubes?

$$a^3+b^3=$$  

• $$\left(a^2+b^2\right)\left(a-b\right)$$  

• $$\left(a+b\right)\left(a-b\right)$$  

• $$\left(a+b\right)\left(a+b\right)\left(a+b\right)$$  

• $$\left(a+b\right)\left(a^2-ab+b^2\right)$$  

Multiple Choice

Which of the following is the rule for factoring the difference of squares?

• $$a^2-b^2=\left(a+b\right)\left(a+b\right)$$  

• $$a^2-b^2=\left(a+b\right)\left(a-b\right)$$  

• $$a^2-b^2=\left(a-b\right)\left(a-b\right)$$  

• $$a^2-b^2=\left(a^2-2ab+b^2\right)$$  

Multiple Choice

Factor:   x³ + 1

• (x + 1)³
• (x+1)(x² - x + 1)
• (x - 1) (x² + x - 1)
• (x + 1)(x² - 2x + 1)

Multiple Choice

Factor: 8x³ - 1

• (2x - 1)(4x² + 2x + 1)
• (2x + 1)(4x² - 2x + 1)
• (2x - 1)(4x² - 2x - 1)
• (2x + 1)(4x² +2x + 1)

Multiple Choice

Noah writes an expression that gives the volume of an open-top box in cubic inches in terms of the length of x in inches of the cutout squares use to make it. The graph that Noah gets is given. What is a functional domain for this graph?

• $$-1<x<5$$

• $$x>0$$

• $$0<x<2.5$$

• $$0<x<3.5$$

Multiple Choice

Noah writes an expression that gives the volume of an open-top box in cubic inches in terms of the length of x in inches of the cutout squares use to make it. The graph that Noah gets is given. What is Noah's approximate maximum value??

• 0

• 10

• 4

• 15