Special Products of Polynomials

Objectives Recognize special polynomial product patterns. Use special polynomial product patterns to multiply two polynomials.

Review Multiplication of polynomials is an extension of the distributive property. When you multiply two polynomials you distribute each term of one polynomial to each term of the other polynomial. We can multiply polynomials in a vertical format like we would multiply two numbers. (x – 3) (x – 2) _________ x –2x + 6 _________ x2 –3x + 0 x2 –5x + 6

Review Multiplication of polynomials is an application of the distributive property. When you multiply two polynomials you distribute each term of one polynomial to each term of the other polynomial. We can also multiply polynomials by using the FOIL pattern. (x – 3)(x – 2) = x(x) + x(–2) + (–3)(x) + (–3)(–2) = x2 – 5x + 6

Special Products Some pairs of binomials have special products. When multiplied, these pairs of binomials always follow the same pattern. By learning to recognize these pairs of binomials, you can use their multiplication patterns to find the product quicker and easier.

Special Products One special pair of binomials is the sum of two numbers times the difference of the same two numbers. Let’s look at the numbers x and 4. The sum of x and 4 can be written (x + 4). The difference of x and 4 can be written (x – 4). Their product is (x + 4)(x – 4) = x2 – 4x + 4x – 16 = x2 – 16 Multiply using foil, then collect like terms.

} Special Products Here are more examples: (x + 4)(x – 4) = x2 – 4x + 4x – 16 = x2 – 16 What do all of these have in common? (x + 3)(x – 3) = x2 – 3x + 3x – 9 = x2 – 9 (5 – y)(5 + y) = 25 +5y – 5y – y2 = 25 – y2

Special Products What do all of these have in common? x2 – 16 x2 – 9 25 – y2 They are all binomials. They are all differences. Both terms are perfect squares.

Special Products For any two numbers a and b, (a + b)(a – b) = a2 – b2. In other words, the sum of two numbers times the difference of those two numbers will always be the difference of the squares of the two numbers. Example: (x + 10)(x – 10) = x2 – 100 (5 – 2)(5 + 2) = 25 – 4 = 21 3 7 = 21

Special Products The other special products are formed by squaring a binomial. (x + 4)2 and (x – 6)2 are two example of binomials that have been squared. Let’s look at the first example: (x + 4)2 (x + 4)2 = (x + 4)(x + 4) = x2 + 4x + 4x + 16 = x2 + 8x + 16 Now we FOIL and collect like terms.

Special Products Whenever we square a binomial like this, the same pattern always occurs. (x + 4)2 = (x + 4)(x + 4) = x2 + 4x + 4x + 16 = x2 + 8x + 16 In the final product it is squared… See the first term? …and it appears in the middle term.

What about the second term? Special Products Whenever we square a binomial like this, the same pattern always occurs. (x + 4)2 = (x + 4)(x + 4) = x2 + 4x + 4x + 16 = x2 + 8x + 16 What about the second term? The middle number is 2 times 4… …and the last term is 4 squared.

Special Products Whenever we square a binomial like this, the same pattern always occurs. (x + 4)2 = (x + 4)(x + 4) = x2 + 4x + 4x + 16 = x2 + 8x + 16 Squaring a binomial will always produce a trinomial whose first and last terms are perfect squares and whose middle term is 2 times the numbers in the binomial, or… For two numbers a and b, (a + b)2 = a2 + 2ab + b2

Special Products Is it the same pattern if we are subtracting, as in the expression (y – 6)2? (y – 6)2 = (y – 6)(y – 6) = y2 – 6y – 6y + 36 = y2 – 12y + 36 It is almost the same. The y is squared, the 6 is squared and the middle term is 2 times 6 times y. However, in this product the middle term is subtracted. This is because we were subtracting in the original binomial. Therefore our rule has only one small change when we subtract. For any two numbers a and b, (a – b)2 = a2 – 2ab + b2

Special Products Examples: (x + 3)2 = (x + 3)(x + 3) Remember: (a + b)2 = a2 + 2ab + b2 = x2 + 2(3)(x) + 32 = x2 + 6x + 9 (z – 4)2 = (z – 4)(z – 4) Remember: (a – b)2 = a2 – 2ab + b2 = z2 – 2(4)(z) + 42 = z2 – 8z + 16

Special Products You should copy these rules into your notes and try to remember them. They will help you work faster and make many problems you solve easier. For any two numbers a and b, (a + b)(a – b) = a2 – b2. For two numbers a and b, (a + b)2 = a2 + 2ab + b2 For any two numbers a and b, (a – b)2 = a2 – 2ab + b2

You Try It. (2x – 5)(2x + 5) (x + 7)2 (x – 2)2 (2x + 3y)2

You Try It. (2x – 5)(2x + 5) (2x – 5)(2x + 5) 22x2 – 52 4x2 – 25

You Try It. 2. (x + 7)2 (x + 7)2 x2 + 2(7)(x) + 72 x2 + 14x + 49

You Try It. 3. (x – 2)2 (x – 2)2 x2 – 2(2)(x) + 22 x2 + 4x + 4

You Try It. 4. (2x + 3y)2 (2x + 3y)2 22x2 – 2(2x)(3y) + 32y2