Math 3 Semester 1 Review (Unit 1)

Unit 1: Polynomials

Polynomial End Behavior

Multiple Choice

What is the end behavior of the polynomial?

• $$As\ x\rightarrow-\infty,\ f\left(x\right)\rightarrow+\infty$$ $$As\ x\rightarrow+\infty,\ f\left(x\right)\rightarrow-\infty$$

• $$As\ x\rightarrow-\infty,\ f\left(x\right)\rightarrow+\infty$$ $$As\ x\rightarrow+\infty,\ f\left(x\right)\rightarrow+\infty$$

• $$As\ x\rightarrow-\infty,\ f\left(x\right)\rightarrow-\infty$$ $$As\ x\rightarrow+\infty,\ f\left(x\right)\rightarrow-\infty$$

Multiple Choice

What is the end behavior of the polynomial?

• $$As\ x\rightarrow-\infty,\ f\left(x\right)\rightarrow-\infty$$ $$As\ x\rightarrow+\infty,\ f\left(x\right)\rightarrow+\infty$$

• $$As\ x\rightarrow-\infty,\ f\left(x\right)\rightarrow+\infty$$ $$As\ x\rightarrow+\infty,\ f\left(x\right)\rightarrow+\infty$$

• $$As\ x\rightarrow-\infty,\ f\left(x\right)\rightarrow-\infty$$ $$As\ x\rightarrow+\infty,\ f\left(x\right)\rightarrow-\infty$$

Polynomial End Behavior

​The degree and leading coefficient of a polynomial can help us find the end behavior of a polynomial's graph.

Polynomials

Multiple Choice

What is the end behavior of the polynomial below? (Hint: Look at the degre and leading coefficient)

$$f\left(x\right)=-2x^3+4x^2-9x+1$$

• $$As\ x\rightarrow-\infty,\ f\left(x\right)\rightarrow-\infty$$ $$As\ x\rightarrow+\infty,\ f\left(x\right)\rightarrow+\infty$$

• $$As\ x\rightarrow-\infty,\ f\left(x\right)\rightarrow+\infty$$ $$As\ x\rightarrow+\infty,\ f\left(x\right)\rightarrow+\infty$$

• $$As\ x\rightarrow-\infty,\ f\left(x\right)\rightarrow+\infty$$ $$As\ x\rightarrow+\infty,\ f\left(x\right)\rightarrow-\infty$$

Domain and Range

Multiple Choice

What is the domain of this function?

• $$-1\le x\le3$$

• $$x<-3$$

• $$-1\le x\le2$$

• $$all\ real\ numbers$$

Multiple Choice

What is the range?

• $$all\ real\ numbers$$  

• $$y\ge-1$$  

• $$-1\le y\le3$$  

• $$y\ge3$$  

Multiple Choice

What is the range?

• $$all\ real\ numbers$$  

• $$y\ge0$$  

• $$0<y<5$$  

• $$y\ge-1$$  

An Absolute Max is the highest point on the graph.

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Absolute Max and Min

​An Absolute Min is the lowest point on the graph.

A relative max is a peak on the graph.
A relative min is a pit on the graph.

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Relative Max and Min

Labelling

Label the following points on the graph.

Drag labels to their correct position on the image

Absolute Max

Relative Min

Absolute Min

Multiple Choice

The blue dot on this graph represents a(n)...

• Absolute Maximum

• Absolute Minimum

• Relative Maximum

• Relative Minimum

Multiple Choice

The point $$\left(1,\ 4\right)$$ is a(n)...

• Absolute Maximum

• Absolute Minimum

• Relative Maximum

• Relative Minimum

Multiple Select

Which interval(s) is the function increasing on?

Select 2 answers.

• $$-2<x<2$$  

• $$x<-1$$  

• $$x>1$$  

• $$-1<x<1$$  

Multiple Choice

Which interval is the function decreasing on?

• $$-2<x<2$$  

• $$x<-1$$  

• $$x>1$$  

• $$-1<x<1$$  

A root or zero, is where the polynomial is equal to zero.

We can find the zeros of a polynomial from its graph by looking at the x -intercepts.

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Zeros of Polynomials

Multiple Choice

What are the "zeros" of this polynomial?

• -2, -1

• -2, -1, 0, 1, 2

• -2, -1, 1, 2

• -2, -1, 4, 1, 2

The zeros of a polynomial have different multiplicities depending on what they look like as they cross the x - axis.

This is helpful for when we are writing the equation of a polynomial in factored form.

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Multiplicities of Zeros

Multiple Choice

Identify a possible equation for the polynomial.

• $$f\left(x\right)=\left(x+3\right)\left(x+1\right)\left(x+5\right)$$

• $$f\left(x\right)=\left(x-3\right)\left(x-1\right)\left(x-5\right)$$

• $$f\left(x\right)=\left(x-3\right)\left(x-1\right)\left(x+5\right)$$

• $$f\left(x\right)=\left(x+3\right)\left(x+1\right)\left(x-5\right)$$

Multiple Choice

Add the polynomials.

$$\left(9x^3+5x^2-3x\right)+\left(4x^3-8x^2+7x\right)$$  

• $$13x^3-3x^2-4x$$  

• $$13x^3-13x^2+4x$$  

• $$13x^3-3x^2+4x$$  

• $$14x^6$$  

Multiple Choice

Subtract the polynomials.

$$(4x^3-3x^2+6x-4)-(-2x^3+x^2-20)$$

• $$4x^2-6x^3-2x+6$$

• $$3x+5x^2+1$$

• $$6x^2-4x^2+6x+16$$

• $$3x+2$$

Multiple Select

Multiply

$$(4x+3)(2x-1)$$

• $$8x^2-2x-3$$

• $$8x^2-3$$

• $$8x^2+6x-3$$

• $$8x^2+2x-3$$

Multiple Select

Multiply

$$(4x+3)(2x-1)$$

• $$8x^2-2x-3$$

• $$8x^2-3$$

• $$8x^2+6x-3$$

• $$8x^2+2x-3$$

​Watch the video below for a review on how to divide Polynomials using Synthetic Division.

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.

If you divide a polynomial by a linear binomial (x- r) and the remainder is zero, then the binomial is a factor.

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Factor Theorem

Multiple Choice

$$\left(2x^5+5x^4-14x^3+17x^2-7x-3\right)\div\left(x-1\right)$$   Is x-1 a factor of the polynomial?

• Yes

• No

Explanation Slide...

Yes, x-1 is a factor of the polynomial because when the polynomial is divided by x-1, the remainder is 0.

Multiple Choice

Is $$(x+2)$$ a factor of $$2x^3+5x^2+9$$  ?

• yes

• no