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A.3
Polynomials and Factoring
In the following polynomial, what is the degree
and leading coefficient?
4x2 - 5x7 - 2 + 3x
Degree
= 7
Leading coef. = -5
Ex.
1
Adding polynomials
(7x4 - x2 - 4x + 2) - (3x4 - 4x2 + 3x)
= 4x4 + 3x2 - 7x + 2
First, dist.
the neg.
Ex.
2
Foil
(3x - 2)(5x + 7) = 15x2 + 11x - 14
Ex.
3
The product of Two Trinomials
(x + y - 2)(x + y + 2) = x2 + xy + 2x + xy + y2 + 2y
-2x - 2y - 4
= x2 + 2xy + y2 - 4
Pascal’s Triangle
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
(a + b)0
(a + b)1
(a + b)2
(a + b)3
(a + b)4
(a + b)5
(a + b)6
Pascal’s Triangle can be used to expand
polynomials that look like....
Ex.
4
Expand
(x + y)3
The row that matches up with this example is row 4.
It is
1 3 3 1
These are the coef. in front
of each term.
1
3
3
1
1x3y0 + 3x2y1 + 3x1y2 + 1x0y3
Notice that the sum of the exponents always add
up to three.
Let’s do (a + b)5
1
5
10
What line of coef. are we
going to use?
10
5
1
a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4 + b5
One more…
(2x - 3y)4
Write down the coef. first.
1
4
6
4
1
a4 + 4a3b + 6a2b2 + 4ab3 + b4
Now let a = 2x and b = -3y
(2x)4 + 4(2x)3(-3y) + 6(2x)2(-3y)2 + 4(2x)(-3y)3 + (-3y)4
16x4 - 96x3y + 216x2y2 - 216xy3 + 81y4
Removing Common Factors
Ex.
5
6x3 - 4x = 2x(3x2 - 2)
(x - 2)(2x) + (x - 2)(3) = (x - 2)(2x + 3)
3 - 12x2 = 3(1 - 4x2)
= 3(1 - 2x)(1 + 2x)
Factoring the Difference of Two Squares
Ex.
6
(x + 2)2 - y2 =
(x + 2 - y)(x + 2 + y) or
(x - y + 2)(x + y + 2)
16x4 - 81 =
(4x2 - 9)(4x2 + 9)
(2x + 3)(2x - 3)(4x2 + 9)
Factoring Perfect Trinomials
Ex.
7
16x2 + 8x + 1 = (4x + 1)(4x + 1)
= (4x + 1)2
Ex.
8
Factor
x2 - 7x + 12
(x - 3)(x - 4)
2x2 + x - 15
(2x - 5)(x + 3)
Factoring the Sum and Difference of Cubes
3
3
3
3
2
2
2
2
u + v = (u + v)(u ! uv + v )
(
u ! v = (u ! v) u + uv + v
)
Ex.
9
x3 - 27 = (x)3 - (3)3
Let u = x
and v = 3
3
3
3
3
Plug these into the diff. of
cubes equation
(
= (x ! 3)(x
2
)
+ 3x + 9)
u ! v = (u ! v) u + uv + v
x !3
2
2
Ex.
10
Factor 3x3 + 192
First, factor out a 3.
3(x3 + 64)
Next, write each term as something cubed and
set them equal to a and b.
3((x)3 + (4)3)
3
3
Let a = x and
b=4
2
2
a + b = (a + b)(a ! ab + b )
2
= 3( x + 4)( x ! 4 x + 16)
Ex.
11
Factoring by Grouping
{
{
x3 - 2x2 - 3x + 6
x2(x - 2) - 3(x - 2)
What can we factor out of
the first two terms? And the
second two terms?
Did you remember to factor
a negative from the +6?
Now what does each group have in common?
Now factor it out.
(x - 2)(x2 - 3)