Multiplying Binomials

Factoring. Lesson #4:

Objectives: Students will explain and use F.O.I.L. to multiply binomials, then use it to derive rules for how to multiply binomials mentally.

NCTM Standards:Students should develop an appreciation of mathematical justification in the study of all mathematical content. In high school, their standards for accepting explanations should become more stringent, and they should develop a repertoire of increasingly sophisticated methods of reasoning and proof.

California Content Standards:10.0 Students add, subtract, multiply, and divide monomials and polynomials.

Can we use the distributive property to multiply a binomial by a binomial?

Multiplying Binomials

We know how to multiply a binomial by a monomial:

Suppose a = (x + 1).

a ( x + 2)

Can we distribute (x + 1) across (x + 2) ? The answer is yes.

First multiply (x + 1) ( x ).

Then multiply (x + 1) ( 2 ) .

(x + 1) ( x + 2) ?

How do we find this product:

(x2 + x) + (2x + 2)

x2 + 3x + 2

= ax + 2a

(x + 1) ( x ) + (x + 1) ( 2 )(x + 1) (x + 2) =

F.O.I.L

(x + 1) (x + 2) = x ( x + 2 ) + 1 ( x + 2 )

If we perform our distribution in this order

First + Outer + Inner + Last

a particularly useful pattern emerges.

(x + 1)(x + 2) = x (x + 2) + 1 (x + 2)

Distributing produces the sum of these four multiplications.

"F.O.I.L" for short.

x2 + 2x + x + 2

x2 + 3x + 2

Multiplying Binomials Mentally

(x + 2)(x + 1)

(x + 3)(x + 2)

(x + 4)(x + 3)

(x + 5)(x + 4)

(x + 6)(x + 5)

x2 + x + 2x + 2

x2 + 2x + 3x + 6

x2 + 3x + 4x + 12

x2 + 4x + 5x + 20

x2 + 5x + 6x + 30 x2 + 11x + 30

x2 + 9x + 20

x2 + 7x + 12

x2 + 5x + 6

x2 + 3x + 2

Later we will use this pattern "in reverse" to factor trinomials that are the product of two binomials.

(x + a)(x + b) = x2 + (a + b) x + ab

There are lots of patterns here, but this one

enables us to multiply binomials mentally.

Can you see a pattern?

Practice: Multiplying Binomials Mentally

1. What is the last term when (x + 3) is multiplied by (x + 6) ?

18 18 = 6 times 3

2. What is the middle term when (x + 5) is multiplied by (x + 7) ?

12x 12 = 5 plus 7

3. Multiply: (x + 4) (x + 7)

4. Multiply: (x + 7) (x + 4)

x2 + 11x + 28 4 plus 7 = 11 4 times 7 = 28

x2 + 11x + 28 7 plus 4 = 11 7 times 4 = 28

Positive and Negative

All of the binomials we have multiplied so far have been sums of positive numbers. What happens if one of the terms is negative?

Example 1:

1. The last term will be negative, because a positive times a negative is negative.2. The middle term in this example will be positive, because 4 + (- 3) = 1.

Example 2:

(x + 4)(x - 3)

1. The last term will still be negative, because a positive times a negative is negative.2. But the middle term in this example will be negative, because (- 4) + 3 = - 1.

(x - 4)(x + 3) = x2 - x - 12

(x + 4)(x - 3) = x2 + x - 12

(x - 4)(x + 3)

Two Negatives

What happens if the second term in both binomials is negative?

Example:

1. The last term will be positive, because a negative times a negative is positive.

2. The middle term will be negative, because a negative plus a negative is negative.

(x - 4)(x - 3)

(x - 4)(x - 3) = x2 -7x +12

Compare this result to what happens when both terms are positive:

(x + 4)(x + 3) = x2 +7x +12

Both signs the same: last term positivemiddle term the same

Sign Summary

(x + 4)(x + 3)

Middle Term Last Term

positive positive

(x - 4)(x + 3) negative negative

(x + 4)(x - 3) positive negative

(x - 4)(x - 3) negative positive

Which term is bigger doesn't matter when both signs are the same, but it does when the signs are different.

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