P.4 Factoring Polynomials

Common FactorsFactor Special Forms

Factor Trinomials w/Binomial Factors

Factor by Grouping

factoring out common factors

• When asked to factor a polynomial, the FIRST thing you should try to do is: PULL OUT ANY COMMON FACTORS

• ab + ac = a(b + c) is the __________property.

• 15x + 9 = __ (__ + __)

• You can also factor out variables:

• 7x3 – 10x2 + x = __(____________)

“Outside and Under”• 7x3 – 10x2 + x Let’s go back and show work on this one.

When you think “factoring something out,” you should think of it as “dividing something out,” because it is undistributing a factor.

If the instructions had told you, “Factor out an x,” you could put an x outside and under, like:

using outside and under can help

using outside and under can help

• Factor (x – 2)(2x) + (x – 2)2(3).

Factoring Special Forms

• MEMORIZE THESE PATTERNS:u2 – v2 = (u + v)(u – v) is “difference of squares”u2 + 2uv + v2 = (u + v)2 is “perfect square trinomial”u2 - 2uv + v2 = (u - v)2

is also “perfect square trinomial”

• So you will be watching for polynomials that might fit one of these patterns:• u2 – v2

• u2 + 2uv + v2

• u2 - 2uv + v2

• Train an alarm to go off in your head to alert you when you have possible u2 or v2

• Start by looking for the terms that could be u2 or v2 so terms that are perfect squares like: 9x2, 4y2, w4, 64x4, or even (x – 2)2

• Notice 9x2 = (3x)2

• and 4y2 = (2y)2

• Notice w4 = (w2)2

• and 64x4 = (8x2)2

• Factor: 9w2 – 4

• When you recognize that this is a “difference of squares,” pull this form from memory:

u2 – v2 = (u + v)(u – v)

Factor x2 – 10x + 25• When you see that your have a trinomial, you look at

the 1st and 3rd terms to find perfect squares (the u2 or the v2)

(x)2 – 10x + (5)2

• Drumroll please…Is the middle term 2uv? In this case, 2(x)(5) = 10x

• YAY. Now we know we are allowed to recall from memory u2 - 2uv + v2 = (u - v)2 and use it.

Factor (x + 2)2 – y2

• It looks like u2 – v2, so fill in the blanks:

• (_____ + ____)(_____ - ____)

• You kind of pretend that the (x + 2) is a package. When I taught high school, I circled (x-2) and called it “egg,” to see it as a unit.

Factor 16x2 + 12x + 9

• First check for any common factors.

• Then look at the first and last terms:

• (4x)2 + 12x + 32

• Drumroll. Is the middle term 2uv?

You try:• Factor 16x2 + 24x + 9

You try:

• Factor 16x4 – 81 = (____)2 – (__) 2

= (____ + ____)(____ - ____)Check this answer: NOT DONE YET!

(___ + ___)(___ + ___)(___ - ___)

Factoring Trinomials

• When you’ll be asked to factor a polynomial, it never tells you how to do it. You have to go through your mental checklist of methods:

• 1.Look for common factors to pull out

• 2.Look for one of those special forms

• 3.See if you can do this following method

Factor 2x2 + x - 15

Consider how we can multiply the “FIRST” (in FOIL) terms to get 2x2.

(2x )(1x )

See how the “FIRST” arrow will give you the new term of 2x2?

What two factors make -15?

It would be some combination of either three and five or one and fifteen, where one of them is negative and one is positive.

( -3)( 5)

( 3)( -5)

( 1)( -15)

( -1)( 15)

See how this “LAST” arrow would make -15?

Now the “INNER” and “OUTER”

• Now we just have to figure out which combination would give us a middle term of 1x. Try to find it mentally:

• (2x – 1)(x + 15) (2x + 1)(x – 15)

• (2x – 3)(x + 5) (2x + 3)(x – 5)

• (2x – 5)(x + 3) (2x + 5)(x – 3)

• (2x – 15)(x + 1) (2x +15)(x – 1)

• When you have found the combination that works, that’s the answer.

Try to factor: 6x2 – 2x – 20

• Practice, practice, practice!

Factoring by Grouping• Factor x3 – 2x2 – 3x + 6.• With grouping, we want to put in parentheses,

but we need to be careful to honor the associative property of addition when we do this. I’d like to “group” the first two terms and “group the last two terms, but there must be a plus sign in the middle because associative property works with addition but not with subtraction.

• Rewrite x3 – 2x2 – 3x + 6 as x3 – 2x2 + -3x + 6.• Now I am allowed to “group” with parentheses,

like (x3 – 2x2) + (-3x + 6)

(x3 – 2x2) + (-3x + 6)

• Now you will look for common factors to pull out of each group:

• ____(________) + ____(_________)

• We need the () parts to be the same so that we can factor them out next!

(x3 – 2x2) + (-3x + 6)

• x2(________) + -3(_________)

• Now look at this. See any common factors to pull out of the whole thing?

• YES! Pull out (x – 2). outside and under

• (x – 2)( )

Sometimes…

• Every once in a while you get a problem that seems to resist grouping, but sometimes the key is to rearrange the factors in the beginning. For example,

• 6 –x4 + 2x – 3x3

• (6 – x4) + (2x – 3x3), but no common factors to pull out of the (6 – x4), BUT

• Rearranged: 6 + 2x – 3x3 – x4 Try it!

6 + 2x – 3x3 – x4