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Chapter 4
Polynomials
Exponents







20= 1
21=2= 2
22=2*2= 4
23=2*2*2= 8
24=2*2*2*2= 16
25=2*2*2*2*2= 32
28=256
Special Cases
1. Anything to the zero
power is 1 x0 = 1
2. Second power is also
called “squared”
3. Third power is also
called “cubed”
Monomial
A monomial is an
expression that is a
number (constant),
variable or the product of
a number and one or more
variables.
x2
x is the base, 2 is the power
Examples

1.
2.
3.
4.
5.
6.
5
X
5x
y3
5x3
5x3y2
Combining Like terms
Like Terms are monomials with the same
variables all raised to the same power.
 You can only add and subtract like terms
Like Terms
Not like Terms
2x + 3x = 5x
2a + 3b
x2 + 3x2 = 4x2
2x2 + 2x3
x2y3 + 2x2y3 = 3x2y3
x2y4 + 2x3y6

Multiplying Monomials
am * an = am+n
x3 * x 5 = x8
x2 * x 3 = x5
2x2 * 3x5 = 6x7
(3n2)(-4n4) = -12n6
1.
2.
3.
Multiply the numbers
(constants)
Take each variable in
alphabetical order
When the bases are the
same add the powers.
Powers of Monomials
1.
2.
3.
Power of a Power
(am)n = amn
(x4)5 = x20
(x3)2 = x6
[(-a)2]3 = (a2)3 = a6
1.
2.
3.
4.
Power of a Product
(ab)m = ambm
(2x)3 = 8x3
(3x2y)2 =9x4y2
(-2a)5 = -32a5
(8x3y4z5)2 = 64x6y8z10
Polynomials


A polynomial is the sum
of monomials.
Polynomials are usually
written with their
variables in alphabetic
order, with highest
powers first.
Examples
1. 2x - 9
2. 3x2 + 2x - 3
3. 4x3y + 3x2y2 - 2xy3 +4
4. 2a + 3b2 – 4c +5d3
Multiplying Polynomials
by Monomials
Distribute to each term, then combine like terms.
1.
x(x+3) = x2 + 3x
2.
3y(2y+4) = 6y2 + 12y
3.
-2x(4x2 - 3x + 5) = -8x3 + 6x2 - 10x
4.
5xy2(3x2 - 4xy + y2) = 15x3y2 - 20x2y3 + 5xy4
Multiplying Polynomials
1.
2.
3.
4.
5.
(x+3)(x+2) = x2
(x+3)(x+2)=x2 + 2x
(x+3)(x+2)= x2 + 2x + 3x
(x+3)(x+2)= x2 + 2x + 3x + 6
(x+3)(x+2)= x2 + 5x + 6
First
Outer
Inner
Last
Simplify
Multiplying Polynomials
(3x-2)(2x2-5x-4) = 6x3
2. (3x-2)(2x2-5x-4) = 6x3-15x2
3. (3x-2)(2x2-5x-4) = 6x3-15x2-12x
4. (3x-2)(2x2-5x-4) = 6x3-15x2-12x-4x2
5. (3x-2)(2x2-5x-4) = 6x3-15x2-12x-4x2 +10x
6. (3x-2)(2x2-5x-4) = 6x3-15x2-12x-4x2+10x+8
6x3-15x2-12x-4x2+10x+8 = 6x3-19x2-12x+10x+8
6x3-19x2-12x+10x+8 = 6x3-19x2-2x+8
6x3-19x2-2x+8
1.
Transforming Formulas
a = ½ bx solve for b
a * 2 = 1/2bx * 2
2a = bx
2a / x = bx / x
2a/x = b
b = 2a/x
c = ax – b solve for x
c + b = ax – b + b
c + b = ax
(c + b) / a = ax / a
(c + b) / a = x
x = (c + b) / a
A Problem Solving Plan
1.
2.
3.
4.
5.
Read the problem carefully, making a sketch
may help.
Choose variables for the facts given.
Reread the problem and write an equation.
Solve the equation
Check your results.
Rate, Time and Distance
D=R*T

Bicyclists Brent and
Jane started at noon
from points 60 km
apart and rode toward
each other, meeting at
1:30pm Brent’s speed
was 4 km/h greater
than Jane’s speed.
Find their speeds.




It took them 1.5 hours to
meet.
If Jane is going ‘j’ fast then
Brent must be going ‘j +4’
They traveled 60 m.
1.5(j+4) + 1.5j = 60
Solving
Jane = 18
Brent = 22
Area Problems
A=L*W
A rectangle is three times as  The original area minus
long as it is wide. If it
36 is equal to the new
length and width are
area.
both decreased by 2,
 L = 3w
then its area is decreased w(3w) – 36 = (w-2)(3w-2)
by 36. Find its original
 Solving we find w = 5,
dimensions.
so l = 15.
Problems with no solution
Not all word problems have solutions. Here are
some reason for this:
1. Not enough information is given.
2. The given fact lead to an unrealistic result.
3. The given facts are contradictory.