# Apply the Remainder and Factor Theorems

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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

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

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

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

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

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. 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

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

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

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).

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).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Find ALL the ZEROs of the polynomial function.
Example )

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..

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

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