4.1 P OLYNOMIAL F UNCTIONS Objectives: Define a polynomial Divide polynomial Apply the Remainder...

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4.1 POLYNOMIAL FUNCTIONS Objectives: • Define a polynomial • Divide polynomial • Apply the Remainder Theorem, the Factor Theorem, and the connections between remainders and factors. • Determine the maximum number of zeros of a polynomial

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Transcript of 4.1 P OLYNOMIAL F UNCTIONS Objectives: Define a polynomial Divide polynomial Apply the Remainder...

Page 1: 4.1 P OLYNOMIAL F UNCTIONS Objectives: Define a polynomial Divide polynomial Apply the Remainder Theorem, the Factor Theorem, and the connections between.

4.1 POLYNOMIAL FUNCTIONSObjectives:•Define a polynomial•Divide polynomial•Apply the Remainder Theorem, the Factor Theorem, and the connections between remainders and factors.•Determine the maximum number of zeros of a polynomial

Page 2: 4.1 P OLYNOMIAL F UNCTIONS Objectives: Define a polynomial Divide polynomial Apply the Remainder Theorem, the Factor Theorem, and the connections between.

POLYNOMIAL FUNCTIONS

Examples Non-examples

•Coefficients, variable, exponents, and constant term•All exponents are whole numbers •No variable on the denominator•No variable under a radical

Page 3: 4.1 P OLYNOMIAL F UNCTIONS Objectives: Define a polynomial Divide polynomial Apply the Remainder Theorem, the Factor Theorem, and the connections between.

DEGREE OF A POLYNOMIAL

What does a first-degree polynomial look like?

What is the degree of the zero polynomial?

POLYNOMIAL FUNCTION

Tell me some polynomial functions that you know!

Page 4: 4.1 P OLYNOMIAL F UNCTIONS Objectives: Define a polynomial Divide polynomial Apply the Remainder Theorem, the Factor Theorem, and the connections between.

POLYNOMIAL DIVISION – JUST LIKE LONG DIVISION.

How does long division work with number?

Notice you must include all powers for the dividend

Page 5: 4.1 P OLYNOMIAL F UNCTIONS Objectives: Define a polynomial Divide polynomial Apply the Remainder Theorem, the Factor Theorem, and the connections between.

YOUR TURN: CHECK ANSWER ON PG. 241

Page 6: 4.1 P OLYNOMIAL F UNCTIONS Objectives: Define a polynomial Divide polynomial Apply the Remainder Theorem, the Factor Theorem, and the connections between.

CHECKING OUR DIVISION.

The Division Algorithm

Remainders and Factors

15 ÷ 3 = 5 and I can check by: 3×5=15

Page 7: 4.1 P OLYNOMIAL F UNCTIONS Objectives: Define a polynomial Divide polynomial Apply the Remainder Theorem, the Factor Theorem, and the connections between.

EXAMPLE 2: FACTORS DETERMINED BY DIVISION

It is a factor if it divides with no remainder! So check:

Page 8: 4.1 P OLYNOMIAL F UNCTIONS Objectives: Define a polynomial Divide polynomial Apply the Remainder Theorem, the Factor Theorem, and the connections between.

EXAMPLE 3: FIND THE REMINDER: IS THE DIVISOR A FACTOR?

Remainder Theorem will help us solve easily!

Read Theorem carefully!!

Page 9: 4.1 P OLYNOMIAL F UNCTIONS Objectives: Define a polynomial Divide polynomial Apply the Remainder Theorem, the Factor Theorem, and the connections between.

WHAT IF THE REMINDER WAS ZERO IN OUR LAST EXAMPLE?:

Page 10: 4.1 P OLYNOMIAL F UNCTIONS Objectives: Define a polynomial Divide polynomial Apply the Remainder Theorem, the Factor Theorem, and the connections between.

EXAMPLE 4: YOUR TURN: THE FACTOR THEOREM

Page 11: 4.1 P OLYNOMIAL F UNCTIONS Objectives: Define a polynomial Divide polynomial Apply the Remainder Theorem, the Factor Theorem, and the connections between.

CONNECTIONS: IF ONE IS TRUE THEN THE OTHERS WILL BE TRUE!!

•-1 and 3 are the zeros of f.•-1 and 3 are the x-intercepts of the graph of f.•-1 and 3 are the solutions to f(x) =0 (x^2 -2x -3 =0)•x +1 and x -3 are factors of x^2 -2x -3

Page 12: 4.1 P OLYNOMIAL F UNCTIONS Objectives: Define a polynomial Divide polynomial Apply the Remainder Theorem, the Factor Theorem, and the connections between.

EXAMPLE 5: A POLYNOMIAL WITH SPECIFIC ZERO

What does the at most mean?

Page 13: 4.1 P OLYNOMIAL F UNCTIONS Objectives: Define a polynomial Divide polynomial Apply the Remainder Theorem, the Factor Theorem, and the connections between.

END SUMMARY

Page 14: 4.1 P OLYNOMIAL F UNCTIONS Objectives: Define a polynomial Divide polynomial Apply the Remainder Theorem, the Factor Theorem, and the connections between.

4.1 HMWR P.248: 1-7ODD, 17-33ODD, 41, 43, 51, 55, 59


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