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

2. 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

3. Divide a polynomial by a polynomial.
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4. Dividing a Polynomial by a Polynomial
EXAMPLE 5
Dividing a Polynomial by a Polynomial
Divide
Solution:
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5. Dividing a Polynomial by a Polynomial
EXAMPLE 6
Dividing a Polynomial by a Polynomial
Divide
Solution:
Remember to include “ ” as part of the answer.
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6. Dividing into a Polynomial with Missing Terms
EXAMPLE 7
Dividing into a Polynomial with Missing Terms
Divide x3 − 8 by x − 2.
Solution:
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7. Your turn
Use polynomial long division to divide
What is the quotient and the remainder
Answer 3x-8,R 0

8. 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

9. 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.

10. 3 9 15 36 3 5 12 34

11. To check your answer use the long division.

12. Your turn
Answers:

13. 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

14. 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