Dividing Polynomials What you’ll learn

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Presentation transcript:

Dividing Polynomials What you’ll learn To divide polynomials using long division. To divide polynomials using synthetic division. Vocabulary Synthetic division

Divide a polynomial by a polynomial. To divide a polynomial by a polynomial (other than a monomial). Both polynomials must first be written in descending powers. Slide 5.7-10

Divide a polynomial by a polynomial. Slide 5.7-11

Dividing a Polynomial by a Polynomial EXAMPLE 5 Dividing a Polynomial by a Polynomial Divide Solution: Slide 5.7-12

Dividing a Polynomial by a Polynomial EXAMPLE 6 Dividing a Polynomial by a Polynomial Divide Solution: Remember to include “ ” as part of the answer. Slide 5.7-13

Dividing into a Polynomial with Missing Terms EXAMPLE 7 Dividing into a Polynomial with Missing Terms Divide x3 − 8 by x − 2. Solution: Slide 5.7-14

Your turn Use polynomial long division to divide What is the quotient and the remainder Answer 3x-8,R 0

Step 1: Use the factor theorem that said: B. Step 1: Use the factor theorem that said: the expression x-a is a factor of a polynomial if and only if the value a is a zero of the related polynomial. Since P(2) =0 then x-2 is a factor of P(x) Step 2:Use the long division to find the others factors So

Take a note There is another way to do the long division and it is called the synthetic division, this method simplifies the long division process by dividing by a linear expression x-a. to use synthetic division, write the coefficients including zeros of the polynomial in standard form. Omit all variables and exponents. For the divisor, reverse the sign (use a). This allows you to add instead of subtract throughout the process.

3 9 15 36 3 5 12 34

To check your answer use the long division.

Your turn Answers:

A theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements. The Remainder theorem: If you divide P(x) of degree by x-a, then is P(a). Here’s why it works when you divide polynomial P(x) by D(x), you find P(x)=D(x)Q(x)+R(x) P(x)=(x-a)Q(x)+R(x) Substitute (x-a) for D(x) P(x)=(a-a)Q(a)+R(a) Evaluate P(a).Substitute a for x =R(a) Simplify

Problem 4: Evaluating a polynomial Given ,What is P(3)? By the remainder theorem P(3) is the remainder when you divide P(x) by x-3 Answer: P(3)=182 Your turn Given that What is P(-4)? Answer: P(-4)=0