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**Products and Factors of Polynomials**

Objective: To multiply polynomials; to divide polynomials by long division and synthetic division.

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**Multiplying Polynomials**

When multiplying two binomials, you FOIL. However, what we are really doing is the distributive property. When multiplying trinomials or larger, all we can do is distribute.

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Example 1 Multiply the following: A B C

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Example 1 Multiply the following: A B C

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Example 1 Multiply the following: A B C

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Example 1 Multiply the following: A B C

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Example 2

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Example 2

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Example 2

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Try This Factor the following polynomials.

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Try This Factor the following polynomials.

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Try This Factor the following polynomials.

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**Sum/Difference of 2 Cubes**

To factor the sum or difference of 2 cubes, you must memorize the following definitions.

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Example 3

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Example 3

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Example 3

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Try This Factor the following polynomials.

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Try This Factor the following polynomials.

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Try This Factor the following polynomials.

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The Factor Theorem The factor theorem states the relationship between the linear factors of a polynomial expression and the terms of the related polynomial functions (x – r) is a factor of the polynomial expression that defines the function P if and only if r is a solution of P(x) = 0, that is, if and only if P(r) = 0.

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Example 4

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Example 4

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Example 4

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Try This Use substitution to determine whether x + 3 is a factor of x3 – 3x2 -6x + 8.

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Try This Use substitution to determine whether x + 3 is a factor of x3 – 3x2 -6x + 8. x + 3 needs to be rewritten as x – (-3), so r = -3.

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Dividing Polynomials A polynomial can be divided by a divisor of the form x – r by using long division or by a method called synthetic division. Long division of polynomials is similar to long division of real numbers. We will follow the same pattern.

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Dividing Polynomials Divide the following:

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**Dividing Polynomials Divide the following: What do we multiply**

x2 by to get x3 ? x.

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**Dividing Polynomials Divide the following: What do we multiply**

x2 by to get x3 ? x. Subtract.

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**Dividing Polynomials Divide the following: What do we multiply**

x2 by to get x3 ? x. Subtract. Bring down the -12.

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**Dividing Polynomials Divide the following: What do we multiply**

x2 by to get x3 ? x. Subtract. Bring down the -12. x2 by to get -2x2 ? -2.

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Example 5

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Example 5

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Try This Find the quotient.

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Try This Find the quotient.

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Synthetic Division With synthetic division, we will only write the coefficients, not the variables. Again, you will need to memorize the following problem solving skill.

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Synthetic Division Divide the following using synthetic division.

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**Synthetic Division Divide the following using synthetic division.**

2 |

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**Synthetic Division Divide the following using synthetic division.**

2 | 1

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**Synthetic Division Divide the following using synthetic division.**

2 | 2 1 5

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**Synthetic Division Divide the following using synthetic division.**

2 |

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**Synthetic Division Divide the following using synthetic division.**

2 |

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**Synthetic Division Divide the following using synthetic division.**

2 | The answer is

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Try This Divide the following using synthetic division.

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**Try This Divide the following using synthetic division. -3 | 1 2 3 8**

-3 | The answer is

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Remainder If there is a nonzero remainder after dividing with either method, it is usually written as the numerator of a fraction, with the divisor as the denominator. Also notice how every power of x needs to be there. Divide

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Remainder If there is a nonzero remainder after dividing with either method, it is usually written as the numerator of a fraction, with the divisor as the denominator. Also notice how every power of x needs to be there. Divide -3|

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Remainder If there is a nonzero remainder after dividing with either method, it is usually written as the numerator of a fraction, with the divisor as the denominator. Also notice how every power of x needs to be there. Divide -3|

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Example 6

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Example 6

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Example 7

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Example 7

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Try This Given

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Try This Given 3|

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Try This Given 3|

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Homework Pages 15-96 multiples of 3 Skip 33

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Division and Factors When we divide one polynomial by another, we obtain a quotient and a remainder. If the remainder is 0, then the divisor is a factor.

Division and Factors When we divide one polynomial by another, we obtain a quotient and a remainder. If the remainder is 0, then the divisor is a factor.

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