9 6 Special Products(3)

8/19/2019 9 6 Special Products(3)

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1. Recognize special polynomial product patterns.

2. Use special polynomial product patterns to multiply

two polynomials.


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1. (x7)(16x2)=Simpliy t!e ollowing"

2. (#$x2y2)(#2xy)(%y$)=

$. (#$a2)2(&a% '$)$=

%. #$x2(2x ) =

&. (%x 7y) (x * &y)=


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+ultiplication o polynomials is an extension o t!e distri'uti,e

property. -!en you multiply two polynomials you distri'ute eac!

term o one polynomial to eac! term o t!e ot!er polynomial.

-e can multiply polynomials in a ,ertical ormat lie we would

multiply two num'ers.(x * $)(x * 2) /////////

6 *2x

0 *$xx2 ///////// x2 *&x 6


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+ultiplication o polynomials is an application o t!e distri'uti,e

property. -!en you multiply two polynomials you distri'ute eac!

term o one polynomial to eac! term o t!e ot!er polynomial.

-e can also multiply polynomials 'y using t!e 34 pattern.

(x * $)(x * 2) = x2

* &x 6x(x) x(*2) (*$)(x) (*$)(*2) =


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Some pairs o 'inomials !a,e special products.

-!en multiplied5 t!ese pairs o 'inomials always ollow t!e

same pattern.

y learning to recognize t!ese pairs o 'inomials5 you can use

t!eir multiplication patterns to ind t!e product uicer and

easier.


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ne special pair o 'inomials is t!e sum o two num'ers times

t!e dierence o t!e same two num'ers.

4et8s loo at t!e num'ers x and %. 9!e sum o x and % can 'ewritten (x %). 9!e dierence o x and % can 'e written (x * %).

9!eir product is

(x %)(x * %) =

+ultiply using oil5 t!en collect lie terms.

x2 * %x %x * 16 = x2 * 16


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(x %)(x * %) = x2 * %x %x * 16 = x2 * 16

:ere are more examples"

(x $)(x * $) = x2 * $x $x * ; = x2 * ;

(& * y)(& y) = 2& &y * &y * y2 = 2& * y2


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x2 * 16 x2 * ; 2& * y2

-!at do all o t!ese

!a,e in common

9!ey are all 'inomials.

9!ey are all dierences.

ot! terms are perect suares.


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or any two num'ers a and b5 (a b)(a * b) = a2 * b2.

3n ot!er words5 t!e sum o two num'ers times t!e dierence o

t!ose two num'ers will always 'e t!e dierence o t!e suares ot!e two num'ers.

>xample" (x 10)(x * 10) = x2 * 100

(& * 2)(& 2) = 2& * % = 21 $ 7 = 21


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9!e ot!er special products are ormed 'y suaring a 'inomial.

(x %)2 and (x * 6)2 are two example o 'inomials t!at !a,e 'een

suared.

4et8s loo at t!e irst example" (x %)2

(x %)2 = (x %)(x %) =

?ow we 34 and collect lie terms.

x2 %x 16 = %x x2 x 16


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(x %)2

= (x %)(x %) = x2

%x 16 = %x x2

x 16

-!ene,er we suare a 'inomial lie t!is5 t!e same pattern always occurs.

See t!e

irst term

3n t!e inal product

it is suared@

@and it appears in t!e middle term.


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(x %)2

= (x %)(x %) = x2

%x 16 = %x x2

x 16


-!at a'out t!e

second term

@and t!e last term is %

suared.

9!e middle num'er is 2 times %@


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(x %)2

= (x %)(x %) = x2

%x 16 = %x x2

x 16


Suaring a 'inomial will always produce a trinomial w!ose irst

and last terms are perect suares and w!ose middle term is 2

times t!e num'ers in t!e 'inomial5 or@

or two num'ers a and b5 (a b)2 = a2 2ab b2


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3s it t!e same pattern i we are su'tracting5 as in t!e expression

(y * 6)2

(y * 6)2

= (y * 6)(y * 6) = y2

* 6y $6 = * 6y y2

* 12y $6

3t is almost t!e same. 9!e y is suared5 t!e 6 is suared and t!e

middle term is 2 times 6 times y. :owe,er5 in t!is product t!e

middle term is su'tracted. 9!is is 'ecause we were su'tracting in t!e

original 'inomial. 9!ereore our rule !as only one small c!angew!en we su'tract.

or any two num'ers a and '5 (a – b)2 = a2 * 2ab b2


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>xamples"

(x $)2 = (x $)(x $) Remem'er" (a + b)2 = a2 + 2ab + b2

= x2 2($)(x) $2

= x2 6x ;

(z * %)2 = Remem'er" (a – b)2 = a2 – 2ab + b2(z * %)(z * %)

= z2 * 2(%)(z) %2

= z2 * z 16


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Aou s!ould copy t!ese rules into your notes and try to remem'er

t!em. 9!ey will !elp you wor aster and mae many pro'lems you

sol,e easier.

or any two num'ers a and '5 (a – b)2 = a2 * 2ab b2

or two num'ers a and b5 (a b)2 = a2 2ab b2

or any two num'ers a and b5 (a b)(a * b) = a2 * b2.


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1. (2x * &)(2x &)2. (x 7)2

$. (x * 2)2

%. (2x $y)2


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1. (2x * &)(2x &)

(2x * &)(2x &)

22x2 * &2

%x2 * 2&


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(x 7)2

x2 2(7)(x) 72

x2 1%x %;

2. (x 7)2


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(x * 2)2

x2 * 2(2)(x) 22

x2 %x %

$. (x * 2)2


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(2x $y)2

22x2 * 2(2x)($y) $2y2

%x2 12x ;y2

%. (2x $y)2

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9 6 Special Products(3)