1. To divide polynomials using long division.
Dividing Polynomials
What you’ll learn
To divide polynomials using long division.
To divide polynomials using synthetic division.
To use the remainder Theorem of Algebra
Vocabulary
Synthetic division, Remainder Theorem,
Algorithm: is an effective method expressed as
a finite list of well-defined instructions for
calculating a function. Algorithms are used for
calculation, data processing, and automated reasoning.
In simple words an algorithm is a step-by-step procedure
for calculations.

2. You can divide polynomials the same way that you
divide whole numbers.
21 divides into 67
3 times
21 divides into 42
2 times

3. Problem 1: Using Polynomial Long Division
Use polynomial long division to divide
What is the quotient and remainder.
Divide
Multiply
Subtract to get 3x and bring down -16
Remember -
Divide
Multiply
The quotient is with the remainder -31
So, this is the polynomial and
is called divisor.

4. You can say 4x+3, R -31 and you can check it
Your turn
Use polynomial long division to divide
What is the quotient and the remainder
Answer 3x-8,R 0

5. Note: the Division Algorithm for Polynomials
You can divide polynomial P(x) by polynomial D(x) to
get polynomial quotient Q(x) and polynomial remainder
R(x). The result is P(x)=D(x)Q(x) +R(x)
If R(x)=0, then P(x)=D(x)Q(x) and D(x) and Q(x) are factors of P(x). To use long division, P(x) and D(x) should be in standard form with zero coefficients where appropriate. The process stops when the degree of the remainder, R(x) is less than the degree of the divisor, D(x).

6. Problem 2: Checking Factors
Include 0x terms
Remember -
Remember -
The degree of the remainder
is less than the degree of the divisor. STOP!
Remember -
The remainder is not a factor of

7. 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

8. Your turn
Answer:

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. Your turn again
The polynomial expresses the volume,
in cubic inches, of the shadow box shown.
What are the dimensions of the box?
B.If the width of the box is 15 inches, what are the
other two dimensions?
Answer: w=10inches
l=18 inches
h=4 inches

14. 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

15. 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

16. hahahahahhh
Classwork odd Homework even
TB pgs 308,309 Exercises and 44-62