FUN WITH FACTORING (MAT Yes, factoring can be fun I__~ you spend the time needed to master it. Remember, it is like a puzzle - your job is to get all the pieces (factors) to fil~ correctly.

TE~S - Things added or subtracted from each other. FACTORS - Things multiplied together. Factoring is the process of turning terms into factors.

# 1 step in factoring is to look for a GCF !!! If an expression has a greatest common factor and you "pull it out" then that expression is not prime. It may or may not factor further, but you do, not say it is prime if there is a GCF. When the directions are "factor out the GCF~, it means to write it down and put behind it in parentheses the remaining factors. Remember to distribute it back in to check your answer. EXA_MPLE: 6a~b~ _ 9a~b + 12aZb~ -3aZb = 3a~b(2ab - 3a + 4b~ -1) When factoring four terms, after making sure there is not one GCF for all four, then tryfactoring by grouping. After factoring out the GCF from the first two terms and factoring out the GCF from the second two terms, the "bundles" that have been created should be the same. If not, check to be sure you ".pulled out" the GCF’s correctly. Once the "bundles" are the same, it can then be thought of as the GCF and can be brought out front AS A SINGLE FACTI)R. The GCF’s from the first step create the second "bundle"

of terms. *** HINT: If the third term is negative, then when you factor out the second GCF, always pull it out as a negative GCF. Remember that the signs will reverse when you factor out a negative GCF. EXAMPLE:

I

negative ]

When factoring trinomialsby trial and error, it is very useful to observe the signs:

If the third term in the trinomial is a POSITIVE, then the two factors will have the same signs AND they will be whatever sign the second term Of the trinomial is.

Ax2+Bx+C.= ( + )( + )

EXAMPLE: Ax2_Bx+c = ( - )( - )

If the third term in the trinomial is a NEGATIVE, then the two factors will have opposite signs.

~2+Bx-C = ( + )(-)

EDtAMPLE: Ax~ _Bx.- C= ( + )( - )

****REMEMBER AS YOU ARE TRYING THE DIFFERENT cOMBINATIONS, DO NOT WASTE YOUR TIME BY PUTTING TOGETHER IN ONE BUNDLE NUMBERS THAT HAVE A

When factoring a trinomial, it is often helpful to first see if you have a PERFECT SQUARE TRINOMIAL. They have ALL of the following properties:

Ax

f

Bx

Must be a Perfect Square

+

C Must be a Perfect Square

~

Must be twice the Product of the Square Roots of the first and last terms

Must be a Positive

In this course we learn how to factor binomials that are the and DIFFERENCE OF PERFECT SOUAREf~ , SUM OF PERFECT CUBE,S , F~. Both the squares and cubes factor easily and quickly using the following patterns: IDIFFERENCE OF PERFECT SQUARES:I A2 - 82 = (A + B)(A - B) ’

THE SUM OF PERFECT SQUARES DOES NOT FACTOR. (That is of course ailer looking for a GCF. If it has a GCF, pull it out.)

EXAMPLES:,

9m2 + 4 = prime. 27m2+ 12 = 3(9m2 + 4).

A~ +B~ =(A,+B)(A~-AB+B2)

ISUM OF PERFECT CUBES: [DIFFERENCE OF PERFECT CUBES:]

A~-B~=(A-B)(A~ +AB+B2)

These two patterns for cubes take some time to memorize, but they are not difficult. Write the two terms very cleariy as cubes. So if one of the cubes is 27x6 make it read The binomial (first factor or bundle} is simplythe two terms without their cubes attached. The trinomial {second factor or bundle) COMES DIRECTLY FROM THE FIRST BINOMIAL FACTOR you just copied down. Don’t use the original terms to create this trinomial. The trinomial is created as follows:

first term of the binomial, squared.

the product of the binomial t~rms.

~

last term of the omial, squared.

always a positive

Be sure that once you have factored an expression once, that there are no difference of perfect squares or sum/difference of perfect cubes lurking around. F_DGaxMPLES: 16p4- 81 = (4p2~+ 9)(4p2 - 9) = (4p2 + 9)(2p + 3)(2p- 3). Also, if there is a choice to either factor something as a cube OR a square, try to do it as a square first. EZKAMPLES:

x6 _ 64y6 = (x3~ _ (8y3~

=(x -8y (x ÷

difference of squares diff. of cubes AND sum of cubes

~ 2y~xz- 2xy + 4yz) = (x 2y~xz + 2xy + 4yZ)(x+ Sometimes it is necessary to first substitute a variable in for a repeated expression, just to make it easier to factor. F_,XAMPLE:

2(m+ 3)z - 5(m+ 3) - 7 = 2xz - 5x - 7 = (2x~7)(x%

Let x = (m+3) **Dod¢ forgetto put(m÷3) backin

= (?.(m+3> - 7) ((m+3) + 1) = C2m+6-7)Cm+4) = (2m-1)(m+4) One last comment about vocabulary - when we say that an expression is ~prime~ or "does not factor~, we are not being completely honest. The fact of the matter is that everything will factor, BUT not within the set of real numbers that we are accustomed to working with. Later, in the next course, you will learn how expressions will factor using imaginary numbers. REMEMBER:

The only way to master a skill is to practice, practice, practice!

Factor out the greatest common factor. 15x’~ya -10x3y2 + 15xTy4 Factor. 2. 2X2 + 4x + 5x + 10 3. 3x7 - 12x5 + 2x3 - 8x Factor the trinomial completely. 4. x2 -7x+12 5. w2 + 3w- 28 6. 3x~ - 3x2 - 6x 7. 8m2 - 10m- 3 8. 6k2 +k-12 Factor completely. 9. 6j4 + 7j2 - 3 ¯ 10. 20e4-51e2

+28

Use the difference of two squares formula to factor the polynomial. 11. 49r2 - 4s2 12. 16d2-9e2 13. -2x4 +32x2 Use the perfect square trinomial formula to factor ~e polynomial. 14. 16f2 +24f+9 15. 9r2 -30r +25 Factor using the sum or difference of two cubes formula. 16. 27m3 - 125 17~ 64aa +27 18. j~-8 19. xa +216 20. Factor the polynomial completely. 20xa - 16x2 ~- 5x + 4,

5xZy~ (3xy - 2 + 3x4y2)

[~] (2x+5)(x+2) [3] x(3x~ +2)(x-2)(x+2) [4] (x- 3)(x-4) [5] (w-4)(w+7) [6] 3x(x ÷ l)(x-2 )

[7] (4m ÷ 1)(2m- 3) [8] (2k ÷ 3)(3k- 4) [9] (3ja - 1)(2jz + 3) [10] (5e~ - 4)(4e~ - 7) [11] (7r + 2s)(7r- 2s) [12] (4d + 3e)(4d- 3e)

[~3] - 2x~ (x + 4)(x-4) [14] (4f + 3)2 [15] (3r -5)~ [16] (3m-5)(9m~ +15m+25) [17] (4a +3)(16aa -12a+9) [18] (j-2)(j= +2j+4) [19] (x + 6)(x~ - 6x + 36)

[20] (2x + 1)(2x - 1)(5x.- 4)