Sections 7.3, 7.4
SPECIAL PRODUCTS OF
POLYNOMIALS,
PROPERTIES OF EXPONENTS
click for printing: 6 slides per page
Product of binomials
 F.O.I.L. method
 Label
L b l tterms
 Firsts, Lasts, Outsides, Insides
 F L F L
(x+3)(x+5)

 O I
I
O
 Multiply:
M li l d
draw li
line, write
i product,
d
d
draw, write…
i




Firsts x2
Outsides +5x
Insides +3x
Lasts +15
 Combine
C bi lik
like tterms: x2+8x+15
8 15
Products of higher degree
polynomials
 Be
B very methodical
th di l
 Draw line for the product of two terms
 Write the product of those terms
 Draw another line for product of terms
 Write the product of those two terms
 Etc: DO NOT DRAW ALL THE LINES AND GO
BACK TO FIND THE PRODUCTS!!
 YOU WILL GET LOST!!
Powers of polynomials
 ((x+3)
3)2 :
 exponent is 2: raised to second power
 (x+3)(x+3) write this way
 Now use F.O.I.L. method
 x2 + 3x + 3x + 9 = x2 + 6x + 9
 x2+3
32 is not the same as (x
(x+3)
3)2
 Put in numbers to test this!!
Shortcut to squaring binomial
 ((x+3)
3)2
 Square first term x2
 Multiply terms together and double it 6x
 Square last term 9
 = x2 + 6x + 9
Product of conjugate factors
 (x+3)(x-3)
( 3)( 3)
 Use FOIL method
 x2 - 3x + 3x - 9 =
 x2 – 9
 Special product: difference of 2 squares
Quadratic function
 Standard
St d d fform
 f(x) = Ax2 + Bx + C
 If it looks different, you need to write it
in this form to use techniques to solve it
Write in standard form:
f(x) = Ax2 + Bx + C
 f(x)
f( ) = -3(x-4)
3( 4)2 + 8
 Square the binomial:(x-4)2=x2-8x+16
 Distribute the -3
2+24x-48
 -3(x
( 2-8x+16)=-3x
)
 Then add 8
 3x2+24x-48+8=
24x 48 8 3x2+24x-40
24x 40
 A=3, B=24, C=-40
Properties of exponents
 These
Th
apply
l to
t factors
f t
off tterms
 NOTE: when there are sums involved, you
need
d to use the
h FOIL method
h d to d
deall with
ih
the terms in parentheses
Product of same variable with
exponents
 x2·x3=(x·x)(x·x·x)=x
( )(
) 5
 x2·x3=x(2+3)=x5
 (2x3y6)(3x8y7)
 =(2·3)(x
( )( 3x8)(
)(y6y7)
 =6x11y13
Raising monomial to a power
 ((-2xy)
2 )3
 =(-2xy)(-2xy)(-2xy) Write out
 =(-2)3x3y3 or cube each term
 =-8x3y3
Quotients with exponents
 x5 = xxxxx
 x2 = xx
 x5 = x(5-2)= x3
 x2
= xxx = x3
Quotients with exponents
 x2 = xx
 x5 = xxxxx
=1
x3
 x2 = x(2-5)= x-3
 x5
 Negative exponents are fractions!!
Quotients with exponents
 x3 = xxx
=1
 x3 = xxx
 x3 = x(3-3)= x0 = 1
 x3
 Zero exponents equal 1!!
How are these different?
 (3
(3+x))2 = 9 + 6x
6 + x2
 (3x)2 = 9x2
 Are they both true?
Raise a power to a power
 (3
(3x3)2
 =(3x3)(3x3) write it out to see steps
 =(3)2(x3x3)
 =9x6
 All exponent rules on page 594