Algebra 1 Glencoe McGraw-Hill JoAnn Evans

7-7 Special Products

Use FOIL to multiply:

(x – 5) (x + 5)

(x + 7) (x - 7)

(a – b) (a + b)

Do you notice a pattern?

(y + 11) (y - 11)

(x – 3) (x + 3)

(n – 1) (n + 1)

Try to put it into words.

x2 - 25 a2 – b2x2 - 49

x2 - 9n2 - 1y2 - 121

The Sum and Difference Pattern

“The sum of any two terms multiplied times the difference of the

same two terms equals the DIFFERENCE of the

SQUARES of the two terms.

(a + b) (a – b)

a2 – ab + ab – b2

The SUM of a and b

times the DIFFERENCE of a and b.

After FOIL, the middle terms cancel because they’re opposites.

a2 – b2 The result is the difference of the squares of the two original terms.

Multiply the binomials using FOIL:

(3n + 7) (3n – 7)

(x + 6y) (x – 6y)

(3a + 2b) (3a – 2b)

(8 + a) (8 – a)(2x - 5) (2x + 5)

Does the Sum and Difference pattern apply here?

9n2 – 21n + 21n – 49

9n2 - 49

4x2 + 10x – 10x – 25

4x2 - 25

64 – 8a + 8a – a2

64 – a2

x2 – 6xy + 6xy – 36y2

x2 – 36y2

9a2 – 6ab + 6ab – 4b2

9a2 – 4b2

Is the answer the square of the original first term minus the square of the original second term?

Definitely!

Use FOIL to multiply the following examples:

(x + 5) (x + 5)

(x + 7) (x + 7)

(a + b) (a + b)

Do you notice a pattern?

(y + 11) (y + 11)

(x + 3) (x + 3)(n + 1) (n + 1)

Try to put it into words.

x2 + 10x + 25

a2 + 2ab + b2

x2 + 14x + 49

x2 + 6x + 9

n2 + 2n + 1

y2 + 22y + 121

“When a binomial is squared (multiplied times

itself), the result is the sum of the squares of the

two terms along with twice their product as the

middle term.” (a + b) (a + b)

a2 + ab + ab + b2

The SUM of a and b times the SUM of a and b.

After FOIL, there are identical middle terms

a2 + 2ab + b2

The result is the square of a, the square of b, and two times the product of a and b.

The Square of a Binomial Pattern

Use FOIL to multiply the following examples:(x - 2) (x - 2) (x - 7) (x - 7) (a - b) (a - b)

Do you notice a pattern?

(y - 10) (y - 10)

(x - 3) (x - 3)(n - 9) (n - 9)

Try to put it into words.

x2 - 4x + 4 a2 - 2ab + b2x2 - 14x + 49

x2 - 6x + 9n2 - 18n + 81

y2 – 20y + 100

“When a binomial is squared (multiplied times

itself), the result is the sum of the squares of the

two terms along with twice their product as the

middle term.”

(a - b) (a - b)

a2 - ab - ab + b2

The DIFFERENCE of a and b times the DIFFERENCE of a and b.

After FOIL, there are identical middle terms

a2 -2ab + b2

The result is the square of a, the square of b, and two times the product of a and b.

The Square of a Binomial Pattern

Write the following problems in exponential form.

(x - 2) (x - 2) (x + 7) (x + 7)

(a - b) (a - b)

(y + 10) (y + 10)

(x - 3) (x - 3)(n + 9) (n + 9)

Recognize the Square of a Binomial in exponential form and in factor form.

(x - 2)2 (a - b)2

(x + 7)2

(x - 3)2(n + 9)2

(y + 10)2

In the beginning of the chapter you learned the Power of a Product rule that said:

To find the power of a product, find the power

of each factor and multiply.

For example: (x • y)2 means x2 • y2What if instead you had (x + 4)2 ?

Inside the parentheses the x and 4 are not factors. They are being added together, not multiplied. The Power of a Product Property doesn’t apply in this case.

The Square of a Binomial

(x + 4)2 DOES NOT MEAN x2 + 42 !

(x + 4)2 DOES NOT MEAN x2 + 16!

(x + 4)2 DOES MEAN (x + 4) (x + 4)

Notice what happens when you multiply using FOIL:

(x + 4) (x + 4) = x2 + 4x + 4x + 16

= x2 + 8x + 16

The Square of a Binomial Pattern

(x + 6)2

The binomial (x + 6) is being squared.

(x + 6) (x + 6)

Expand before multiplying. Use FOIL.

x2 + 6x + 6x + 36

Even though you may remember the pattern, you still need to use FOIL. This will help you INTERNALIZE the pattern so you will recognize them when we begin factoring. x2 + 12x +

36

The Square of a Binomial Pattern

(3n - 5)2 The binomial (3n - 5) is being squared.

(3n - 5) (3n - 5)

Expand before multiplying. Use FOIL.

9n2 – 15n – 15n + 25

Even though you may remember the pattern, you still need to use FOIL. This will help you INTERNALIZE the pattern so you will recognize them when we begin factoring. 9n2 -30n +

25

Some good advice………

Don’t be fooled! (a + b)2 DOES NOT mean a2 + b2

(a + b)2 means (a + b) (a + b)

Use FOIL on this assignment. Yes, you should recognize the patterns that are occurring, but don’t rely on your memory of them. Use FOIL to help your brain internalize the pattern. The recognition of the pattern will be most useful during factoring

Find the product.

210x 1. 7x7x .2 49x2

262x 3. 3624x4x2

225x 4. 420x25x2 9x9x .5 81x2 223x 6. 412x9x2 227x 7. 428x49x2

10020xx2

4x34x3 .8 169x2 292x 9. 8136x4x2 5x125x12 .10 25144x2

7-A8 Pages 407-409 #12–17,20–31,36,61-67.

4x4x.1

5x5x.2 25x2 6x6x.9 36x2

2x22x2.3 44x2 4x34x3.4 169x2

4x24x2.5 164x2

5x45x4.6 2516x2 4x54x5.7 61x52 2 8x68x6.8 64x36 2

5x25x2.10 524x2 2x32x3.11 49x2

7x27x2.12 944x2

3x43x4.13 916x2

8x58x5.14 64x52 2

4x64x6.15 16x36 2

16x2

Use Foil to find the product.

Find the product.

3x33x8.16 9x64 2

Use Foil to find the product.

26x.1 3612xx2 222x.9 48x4x2

232x.2 912x4x2 253x.10 2530x9x2

264x.3 3648x16x2 235x.11 9x03x52 2 227x.4 4x28x49 2 256x.12 52x60x36 2 24x.5 25x.13 2510xx2 26x.6 3612xx2 222x.14 48x4x2 243x.7 1624x9x2 24x2.15 1616x4x2

25x4.16 2540x16x2 22x6.8 4x4236x2

Find the product.

168xx2