9 6 special products

8/13/2019 9 6 Special Products

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1. Recognize special polynomial product patterns.

2. Use special polynomial product patterns to multiply

two polynomials.


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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.(x3)(x2)x_________

+ 62x+ 03xx2_________

x25x + 6


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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.

(x3)(x2) = x2

5x + 6x(x) + x(2) + (3)(x) + (3)(2) =


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Some pairs of binomials havespecial 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.


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One special pair of binomials is the sum of two numbers times

the difference of the same two numbers.

Lets look at the numbers x and 4. The sum of x and 4 can bewritten (x + 4). The difference of x and 4 can be written (x4).

Their product is

(x + 4)(x4) =

Multiply using foil, then collect like terms.

x24x + 4x16 = x216


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(x + 4)(x4) = x24x + 4x16 = x216

Here are more examples:

(x + 3)(x3) = x23x + 3x 9 = x2 9

(5y)(5 + y) = 25 +5y5yy2= 25y2

}What do all of thesehave in common?


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x216 x2 9 25y2

What do all of these

have in common?

They are all binomials.

They are all differences.

Both terms are perfect squares.


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For any two numbers a and b, (a+ b)(ab) = a2b2.

In other words, the sum of two numbers times the difference of

those two numbers will always be the difference of the squares ofthe two numbers.

Example: (x + 10)(x10) = x2100

(52)(5 + 2) = 254 = 213 7 = 21


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The other special products are formed by squaring a binomial.

(x + 4)2and (x6)2are two example of binomials that have been

squared.

Lets look at the first example: (x + 4)2

(x + 4)2 = (x + 4)(x + 4) =

Now we FOIL and collect like terms.

x2+ 4x + 16 =+ 4x x2+ 8x + 16


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(x + 4)2

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

+ 4x + 16 =+ 4x x2

+ 8x + 16

Whenever we square a binomial like this, the same pattern always occurs.

See the

first term?

In the final product

it is squared

and it appears in the middle term.


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(x + 4)2

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

+ 4x + 16 =+ 4x x2

+ 8x + 16


What about the

second term?

and the last term is 4

squared.

The middle number is 2 times 4


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(x + 4)2

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

+ 4x + 16 =+ 4x 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 aand b, (a+ b)2= a2+ 2ab+ b2


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Is it the same pattern if we are subtracting, as in the expression

(y6)2?

(y6)2

= (y6)(y6) = y2

6y + 36 =6y 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 changewhen we subtract.

For any two numbers aand b, (ab)2= a22ab+ b2


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Examples:

(x + 3)2 = (x + 3)(x + 3) Remember: (a + b)2= a2+ 2ab + b2

= x2+ 2(3)(x) + 32

= x2+ 6x + 9

(z4)2= Remember: (a

b)2= a2

2ab + b2(z4)(z4)

= z22(4)(z) + 42

= z28z + 16


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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 aand b, (ab)2= a22ab+ b2

For two numbers aand b, (a+ b)2= a2+ 2ab+ b2

For any two numbers a and b, (a+ b)(ab) = a2b2.


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1. (2x5)(2x + 5)2. (x + 7)2

3. (x2)2

4. (2x + 3y)2


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1. (2x5)(2x + 5)

(2x5)(2x + 5)

22x252

4x225


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(x + 7)2

x2+ 2(7)(x) + 72

x2+ 14x + 49

2. (x + 7)2


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(x2)2

x22(2)(x) + 22

x2+ 4x + 4

3. (x2)2


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(2x + 3y)2

22x22(2x)(3y) + 32y2

4x2+ 12x + 9y2

4. (2x + 3y)2

9 6 special products

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