Algebra 1 Glencoe McGraw-HillJoAnn 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. x a 2 – b 2 x x 2 - 9n 2 - 1y

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) a 2 – ab + ab – b 2 The SUM of a and b times the DIFFERENCE of a and b. After FOIL, the middle terms cancel because they’re opposites. a 2 – b 2 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? 9n 2 – 21n + 21n – 49 9n x x – 10x – 25 4x – 8a + 8a – a 2 64 – a 2 x 2 – 6xy + 6xy – 36y 2 x 2 – 36y 2 9a 2 – 6ab + 6ab – 4b 2 9a 2 – 4b 2 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 + 7) (a + b) Do you notice a pattern? (y + 11) (x + 3) (n + 1) Try to put it into words. x x + 25a 2 + 2ab + b 2 x x + 49 x 2 + 6x + 9n 2 + 2n + 1y y + 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) a 2 + ab + ab + b 2 The SUM of a and b times the SUM of a and b. After FOIL, there are identical middle terms a 2 + 2ab + b 2 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 - 7) (a - b) Do you notice a pattern? (y - 10) (x - 3) (n - 9) Try to put it into words. x 2 - 4x + 4a 2 - 2ab + b 2 x x + 49 x 2 - 6x + 9n n + 81y 2 – 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) a 2 - ab - ab + b 2 The DIFFERENCE of a and b times the DIFFERENCE of a and b. After FOIL, there are identical middle terms a 2 -2ab + b 2 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 + 7) (a - b) (y + 10) (x - 3) (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 x 2 y 2 What 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 x ! (x + 4) 2 DOES NOT MEAN x ! (x + 4) 2 DOES MEAN (x + 4) (x + 4) Notice what happens when you multiply using FOIL: (x + 4) (x + 4) = x 2 + 4x + 4x + 16 = x 2 + 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. x 2 + 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. x x + 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. 9n 2 – 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. 9n 2 -30n + 25

Some good advice……… Don’t be fooled! (a + b) 2 DOES NOT mean a 2 + b 2 (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.

7-A8 Pages #12–17,20–31,36,61-67.

Use Foil to find the product. Find the product.

Use Foil to find the product.Find the product.