3-1 Day 1: Graphing Polynomial Functions

Pearson

Standard form:

Leading coefficient:

Degree:

Polynomial function:

End behavior:

What is each polynomial in standard form and what are the leading coefficient, the degree, and the number of

terms of each?

1. 2𝑥 − 3𝑥4 + 6 − 5𝑥3 2. 𝑥5 + 2𝑥6 − 3𝑥4 − 8𝑥 + 4𝑥3

3. 4𝑥2 + 6𝑥4 − 𝑥7 4. 7 + 𝑥2

Degree is Even Degree is Odd

Leading coefficient is Positive

Example of even degree:

𝑦 = 3𝑥4 − 3𝑥 + 5

Example of odd degree:

𝑦 = 4𝑥5 + 2𝑥

-Rises to left and right

-Falls to left

-Rises to right

Leading coefficient is Negative

Example of even degree:

𝑦 = −3𝑥4 − 3𝑥 + 5

Example of odd degree:

𝑦 = −4𝑥5 + 2𝑥

-Falls to left

-Falls to right

-Rises to left

-Falls to right

Using the leading coefficient and degree of the polynomial function to determine the end behavior of each

graph.

5. 𝑓(𝑥) = 2𝑥6 − 5𝑥5 + 6𝑥4 − 𝑥3 + 4𝑥2 − 𝑥 + 1 6. 𝑔(𝑥) = −5𝑥3 + 8𝑥 + 4

3-1 Day 2: Graphing Polynomial Functions

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Relative minimum:

Relative maximum:

Turning points:

1. Consider the polynomial function 𝑓(𝑥) = 𝑥5 + 18𝑥2 + 10𝑥 + 1.

a. Make a table of values to identify key features and sketch a graph of the function.

b. Find the average rate of change over the interval [0,2].

2. Use the information below to sketch a graph of the polynomial function 𝑦 = 𝑓(𝑥).

f(x) is positive on the intervals (-2, -1) and (1, 2)

f(x) is negative on the intervals (−∞, −2), (-1, 1), and (2, ∞)

f(x) is increasing on the interval (−∞, −1.5)and (0,1.5)

f(x) is decreasing on the intervals (-1.5, 0) and (1.5, ∞)

3-2: Adding, Subtracting, and Multiplying Polynomials

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FOIL:

When subtracting polynomials, don’t forget to subtract __________________________

Add or subtract the polynomials.

1. (4𝑎4 − 6𝑎3 − 3𝑎2 + 𝑎 + 1) + (5𝑎3 + 7𝑎2 + 2𝑎 − 2)

2. (2𝑎2𝑏2 + 3𝑎𝑏2 − 5𝑎2𝑏) − (3𝑎2𝑏2 − 9𝑎2𝑏 + 7𝑎𝑏2)

3. (3𝑥2𝑦2 + 2𝑥𝑦2 + 6𝑥2) − (2𝑥2𝑦2 + 3𝑥𝑦2 − 2𝑥2)

Multiply.

4. (6𝑛 − 7)(𝑛 + 3)

5. (𝑚𝑛 + 1)(𝑚2𝑛 − 1)(𝑚𝑛2 + 2)

3-3 Day 1: Polynomial Identities

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Identity:

Difference of

Squares 𝑎2 − 𝑏2=(𝑎 + 𝑏)(𝑎 − 𝑏) 25𝑥2 − 81= (𝟓𝒙 + 𝟗)(𝟓𝒙 − 𝟗)

Square of a Sum (𝑎 + 𝑏)2 = 𝑎2 + 2𝑎𝑏 + 𝑏2 (3𝑥 + 4𝑦)2 = (3𝑥)2 + 2(3𝑥)(4𝑦) + (4𝑦)2

= 𝟗𝒙𝟐 + 𝟐𝟒𝒙𝒚 + 𝟏𝟔𝒚𝟐

Difference of

Cubes 𝑎3 − 𝑏3 = (𝑎 − 𝑏)(𝑎2 + 𝑎𝑏 + 𝑏2 8𝑚3 − 27 = (2𝑚 − 3)[(2𝑚)2 + (2𝑚)(3) + 32]

= (𝟐𝒎 − 𝟑)(𝟒𝒎𝟐 + 𝟔𝒎 + 𝟗)

Sum of Cubes 𝑎3 + 𝑏3 = (𝑎 + 𝑏)(𝑎2 − 𝑎𝑏 + 𝑏2) 𝑔3 + 64ℎ2 = (𝑔 + 4ℎ)[𝑔2 − (𝑔)(4ℎ) + (4ℎ)2]= (𝒈 + 𝟒𝒉)(𝒈𝟐 − 𝟒𝒈𝒉 + 𝟏𝟔𝒉𝟐)

Use polynomial identities to multiply the expressions.

1. (2𝑥2 + 𝑦3)2 2. (3𝑥2 + 5𝑦3)(3𝑥2 − 5𝑦3)

3. (12 + 15)2 4. 𝑚8 − 9𝑛10

5. 27𝑥9 − 343𝑦6 6. 123 + 8𝑥3

7. (𝑥4 − 81)2

3-3 Day 2: Polynomial Identities

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Pascal’s Triangle:

Patterns in Pascal’s Triangle:

Binomial Theorem:

1. Use Pascal’s Triangle to expand (𝑥 + 𝑦)6

2. Use Pascal’s Triangle to expand (𝑥 + 𝑦)7

3. Use the Binomial Theorem to expand (𝑥 − 1)7

4. Use the Binomial Theorem to expand (2𝑐 + 𝑑)6

3-4 Day 1: Diving Polynomials

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Long Division:

Use long division to divide the polynomials. Then write the dividend in terms of the quotient and remainder.

1. 𝑥3 − 6𝑥2 + 11𝑥 − 6 divided by 𝑥2 − 4𝑥 + 3

2. 16𝑥4 − 85 divided by 4𝑥2 + 9

3. 8𝑥3 + 27 divided by 2𝑥 + 3

3-4 Day 2: Diving Polynomials

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Synthetic Division:

Remainder Theorem:

Use synthetic division.

1. 3𝑥3 − 5𝑥 + 10 by 𝑥 − 1

2. Use synthetic division to show that the remainder of 𝑓(𝑥) = 𝑥3 + 8𝑥2 + 12𝑥 + 5 divided by 𝑥 + 2 is equal

to 𝑓(−2).

3. A technology company uses the function 𝑓(𝑥) = −𝑥3 + 12𝑥2 + 6𝑥 + 80 to model expected annual revenue,

in thousands of dollars, for a new product, where x is the number of years after the product is released. Use the

Remainder Theorem to estimate the revenue in year 5.

4. Use the Remainder Theorem to determine whether the given binomial is a factor of 𝑃(𝑥).

𝑃(𝑥) = 𝑥3 − 10𝑥2 + 28𝑥 − 16; binomial; 𝑥 − 4

3-5: Zeros of Polynomial Functions

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Multiplicity of a zero:

- When the multiplicity of a zero is ______, the

function crosses the x – axis.

- When the multiplicity of a zero if _______, the graph

has a turning point at the x – axis.

Describe the behavior of the graph of the function at each of its zeros.

1. 𝑓(𝑥) = 𝑥(𝑥 + 4)(𝑥 − 1)4 2. 𝑓(𝑥) = (𝑥2 + 9)(𝑥 − 1)5(𝑥 + 2)2

What are the real and complex zeros of the polynomial function shown in the graph? (Better pictures on pg. 164)

3. 𝑓(𝑥) = 2𝑥3 − 8𝑥2 + 9𝑥 − 9

4. What are the solution(s) of the equation 𝑥3 − 7𝑥 + 6 = 𝑥3 + 5𝑥2 − 2𝑥 − 24