1. Apply the Remainder and Factor Theorems
5.5
Apply the Remainder and Factor Theorems
What you should learn:
Goal 1
Divide polynomials and relate the result to the remainder theorem and the factor theorem.
using Long Division
Synthetic Division
Goal 2
Factoring using the “Synthetic Method”
Goal 3
Finding the other ZERO’s when given one of them.
A1.1.5
5.5 The Remainder and Factor Theorem

2. Divide using the long division
ex) x
+ 7
- ( )
- ( )
6.5 The Remainder and Factor Theorem

3. Divide using the long division with Missing Terms
ex)
- ( )
- ( )
- ( )

4. To divide a polynomial by x - c
Synthetic Division
To divide a polynomial by x - c
1. Arrange polynomials in descending powers, with a 0 coefficient for any missing term.
2. Write c for the divisor, x – c. To the right, write the coefficients of the dividend. 3

5. 3
3. Write the leading coefficient of the dividend on the bottom row. 1
4. Multiply c (in this case, 3) times the value just written on the bottom row. Write the product in the next column in the 2nd row. 3 3 1

6. 5. Add the values in the new column, writing the sum in the bottom row.3 3
add 1 7
6. Repeat this series of multiplications and additions until all columns are filled in. 3 3 21
add 1 7 16

7. 7. Use the numbers in the last row to write the quotient and remainder in fractional form.
The degree of the first term of the quotient is one less than the degree of the first term of the dividend.
The final value in this row is the remainder. 3 3 21 48
add 1 7 16 53

8. To divide a polynomial by x - c
Synthetic Division
To divide a polynomial by x - c
Example 1) -1 -1 -3 1 3 -5

9. To divide a polynomial by x - c
Synthetic Division
To divide a polynomial by x - c
Example 2) 2 -2 2 4 1 2 -1 5

10. Factoring a Polynomial
(x + 3)
Example 1)
given that f(-3) = 0. -3 2 11 18 9 -6
-15 -9 2 5 3
multiply
Because f(-3) = 0, you know that (x -(-3)) or (x + 3) is a factor of f(x).

11. Factoring a Polynomial
(x - 2)
Example 2)
given that f(2) = 0. 2 1 -2 -9 18 2
-18 1 -9
multiply
Because f(2) = 0, you know that (x -(2)) or (x - 2) is a factor of f(x).

12. Reflection on the Section
If f(x) is a polynomial that has x – a as a factor, what do you know about the value of f(a)?
assignment

13. Finding Rational Zeros
5.6
Finding Rational Zeros
What you should learn:
Goal 1
Find the rational zeros of a polynomial.
L1.2.1
5.6 Finding Rational Zeros

14. The Rational Zero Theorem
Find the rational zeros of
solution
List the possible rational zeros. The leading coefficient is 1 and the constant term is So, the possible rational zeros are:
5.6 Finding Rational Zeros

15. Find the Rational Zeros of
Example 1)
Find the Rational Zeros of
solution
List the possible rational zeros. The leading coefficient is 2 and the constant term is 30. So, the possible rational zeros are:
Notice that we don’t write the same numbers twice
5.6 Finding Rational Zeros

16. Use Synthetic Division to decide which of the following are zeros of the function 1, -1, 2, -2
Example 2) -2 -2
-10 28 1 5
-14
x = -2, 2
5.6 Finding Rational Zeros

17. Find all the REAL Zeros of the function.
Example 3) 1 5 6 1 1 5 6
x = -2, -3, 1
5.6 Finding Rational Zeros

18. Find all the Real Zeros of the function.
Example 4) 2 6 14 10 2 1 3 7 5 -1 -2 -5 -1 1 2 5
5.6 Finding Rational Zeros

19. -1 -2 -5 -1 1 2 5
x = 2, -1
5.6 Finding Rational Zeros

20. Reflection on the Section
How can you use the graph of a polynomial function to help determine its real roots?
assignment
5.6 Finding Rational Zeros

21. Apply the Fundamental Theorem of Algebra
5.7
Apply the Fundamental Theorem of Algebra
What you should learn:
Goal 1
Use the fundamental theorem of algebra to determine the number of zeros of a polynomial function.
THE FUNDEMENTAL THEOREM OF ALGEBRA
If f(x) is a polynomial of degree n where n > 0, then the equation f(x) = 0 has at least one root in the set of complex numbers.
L2.1.6
5.7 Using the Fundamental Theorem of Algebra

22. Find all the ZEROs of the polynomial function.
Example 1) -5 -5 45 1 -9
x = -5, -3, 3
5.7 Using the Fundamental Theorem of Algebra

23. So, Yes the given x-value
Decide whether the given x-value is a zero of the function.
, x = -5
Example 1) -5 -5 -5 1 1
So, Yes the given x-value
is a zero of the function.
5.7 Using the Fundamental Theorem of Algebra

24. Write a polynomial function of least degree that has real coefficients, the given zeros, and a leading coefficient of 1.
-4, 1, 5
Example 1)
5.7 Using the Fundamental Theorem of Algebra

25. QUADRATIC FORMULA

26. x = 2.732 x = -.732 Find ALL the ZEROs of the polynomial function.
Example )
x = 2.732
x = -.732

27. Find ALL the ZEROs of the polynomial function.
Example #24)
Doesn’t FCTPOLY…Now what?

28. Find ALL the ZEROs of the polynomial function.
Example )

29. Find ALL the ZEROs of the polynomial function.
Example ) -1 -1 5 -9 -1 14 1 -5 9 1
-14
Graph this one….find one of the zeros..

30. Reflection on the Section
How can you tell from the factored form of a polynomial function whether the function has a repeated zero?
At least one of the factors will occur more than once.
assignment