Graphing Polynomials

More Polynomial Investigation

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Review: Roots and Zeros

The roots of a polynomial function, p(x), are the solutions of the equation p(x)=0. Another name for the roots of a function is zeros of a function because at each root, the value of the function is zero. The real roots (or zeros) of a function have the same value as the x-values of the x-intercepts of its graph because the -intercepts are the points where the y-value of the function is zero.

Sometimes roots can be found by factoring and solving for p(x)=0.

Multiple Choice

How many zeros does this polynomial have?

• 0

• 2

• 3

• 5

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

Multiple Choice

Identify a possible equation.

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

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

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

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

Multiple Choice

Pictured is a third degree polynomial. What do you think is the maximum number of roots it can have?

• 0

• 2

• 3

• 4

Can a polynomial of n degree have fewer than n number of roots?

Fill in the Blanks

What do you think is the maximum number of roots a polynomial of degree n can have?

Multiple Choice

What is the degree of the polynomial function
P(x)=3x⁴-7x²-2x⁷-x+4?

• 4
• 7
• 2
• 1

Multiple Choice

What is the minimum degree this polynomial can have?

• 4

• 1

• 2

• 5

Multiple Choice

Identify a possible equation.

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

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

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

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

Multiple Choice

Is the leading coefficient positive or negative

• positive
• negative

Multiple Choice

What is the smallest possible degree of the function

• 1st
• 3rd
• 4th
• 5th

Multiple Choice

In the above graph complete the following end behavior:
As x --> -∞, f(x) --> ____
As x --> +∞, f(x) --> ____

• -∞
  -∞
• +∞
  -∞
• -∞
  +∞
• +∞
  +∞

Even and Odd Degree Polynomials

• EVEN DEGREE ---Both arrows are in the same direction. They point up if the leading coefficient is positive and down id the leading coefficient is negative.

• ODD DEGREE--- Arrows are having opposite direction. One is pointing up and one is pointing down. The lead coefficient is positive if the graph is going up if you trace them from left to right.

Positive/Odd

ODD/NEGATIVE

When the y-values of a graph get very large as the x-values get large, the graph has positive orientation. When the y-values of a graph get very small as the x-values get large, the graph has negative orientation.

Positive/Even

Multiple Select

Check all the even degree polynomials:

Multiple Choice

What is the leading coefficient?

• Positive 
• Negative
• Odd 
• Even

Multiple Choice

What is the leading coefficient?

• Positive
• Negative
• Odd
• Even

Multiple Choice

What is the degree?

• Positive 
• Negative
• Odd 
• Even

You can use a number line to represent the -values for which a polynomial graph is above or below the x-axis. The bold parts of each number line below show where the output values of a polynomial function are positive. That is, where the graph is above the x-axis. The open circles show locations of the -intercepts or roots of the function. Where there is no shading, the value of the function is negative. Write a possible equation.

Triple Root

You can use a number line to represent the -values for which a polynomial graph is above or below the x-axis. The bold parts of each number line below show where the output values of a polynomial function are positive. That is, where the graph is above the x-axis. The open circles show locations of the -intercepts or roots of the function. Where there is no shading, the value of the function is negative. Write a possible equation. 

Open Ended

Write a possible equation for the polynomial described by the number line.

Open Ended

Write a possible equation for the polynomial described by the number line.