# Products and Factors of Polynomials

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Products and Factors of Polynomials
Objective: To multiply polynomials; to divide polynomials by long division and synthetic division.

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.

Example 1 Multiply the following: A B C

Example 1 Multiply the following: A B C

Example 1 Multiply the following: A B C

Example 1 Multiply the following: A B C

Example 2

Example 2

Example 2

Try This Factor the following polynomials.

Try This Factor the following polynomials.

Try This Factor the following polynomials.

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

Example 3

Example 3

Example 3

Try This Factor the following polynomials.

Try This Factor the following polynomials.

Try This Factor the following polynomials.

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.

Example 4

Example 4

Example 4

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

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.

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.

Dividing Polynomials Divide the following:

Dividing Polynomials Divide the following: What do we multiply
x2 by to get x3 ? x.

Dividing Polynomials Divide the following: What do we multiply
x2 by to get x3 ? x. Subtract.

Dividing Polynomials Divide the following: What do we multiply
x2 by to get x3 ? x. Subtract. Bring down the -12.

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.

Example 5

Example 5

Try This Find the quotient.

Try This Find the quotient.

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.

Synthetic Division Divide the following using synthetic division.

Synthetic Division Divide the following using synthetic division.
2 |

Synthetic Division Divide the following using synthetic division.
2 | 1

Synthetic Division Divide the following using synthetic division.
2 | 2 1 5

Synthetic Division Divide the following using synthetic division.
2 |

Synthetic Division Divide the following using synthetic division.
2 |

Synthetic Division Divide the following using synthetic division.

Try This Divide the following using synthetic division.

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

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

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|

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|

Example 6

Example 6

Example 7

Example 7

Try This Given

Try This Given 3|

Try This Given 3|

Homework Pages 15-96 multiples of 3 Skip 33

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