1. Look for a Greatest Common Factor (GCF). The largest factor that will go into every term of your polynomial.
3k4 - 15k7 + 24k9 = 3k4(1 - 5k3 + 8k5)
The GCF can be more than a single term.r(3x + 2) - 8s(3x + 2) = (3x + 2) (r - 8s)
* Remember that the GCF must be part of your final factored form.
2. Determine if your remaining polynomial can be factored further.
3. It is possible, and sometimes necessary, to change the signs
of a polynomial by factoring out a negative factor rather than a positive
one.
-22x - 11 = -11(2x + 1)
Remember that by FOIL,
1. Once the GCF has been taken out, or if there is none, make sure the remaining polynomial is in descending or ascending order of powers.
(x + 2)(3x - 1)
F O I L
= 3x2 - x + 6x - 2
= 3x2 + 5x - 2
F O+I L
d2 + 9d + 8 (descending)
8 + 9d + d2 (ascending)
* Hint: if you have something of the form:
a) -12 - a + a2, rewrite as a2 - a - 12
b) -x2 - 5x - 6, factor out -1: -1(x2 + 5x + 6)
c) -l0h3 + 29h2 + 3h, factor out -h: -h(l0h2 - 29h - 3)
2. Factoring by trial and error may now begin.
a) Is the numerical coefficient of the first term 1?YES z4 + 7z2 - 30
c) Write down factors of last term. 30*1, 15*2, 6*5, 10*3
d) Check the sign of the last term.
1) If positive then signs will be either 2 positives or 2 negatives.e) Check the sign of the middle term and use this with sign of last term to determine signs that will be used.2) If negative, signs will be a positive and a negative.
1) Last positive and middle positive, 2 positives.f) Put parentheses in place and put variables in appropriate positions.2) Last positive and middle negative, 2 negatives.
3) Last negative and middle positive, different signs with larger middle term (0 + I of FOIL) when you multiply being positive.
Last negative and middle negative, different signs with larger middle term when you multiply being negative.
g) Use combination of factors of the last to get correct form which equals original trinomial when multiplied.
NO 18a2 - 3ab - 28b2
b) Write down the factors of the numerical coefficient of first term. 18*1, 9*2, 6*3c) Write down factors of last term. 28*1, 14*2, 7*4
d) Check the sign of the last term.
1) If positive then signs will be either 2 positives or 2 negatives.e) Check the sign of the middle term and use this with sign of last term to determine signs that will be used.2) If negative, signs will be a positive and a negative.
1) Last positive and middle positive, 2 positives.f) Put parentheses in place and put variables in appropriate positions.2) Last positive and middle negative, 2 negatives.
3) Last negative and middle positive, different signs with larger middle term (0 + I of FOIL) when you multiply being positive.
4)Last negative and middle negative, different signs with larger middle term when you multiply being negative.
g) Use combination of factors of 1st term with those of last to get correct form which equals original trinomial when multiplied.
| 2m3 + 30m2
+ 112m
1. 2m(m2 + 15m + 56) 2. a) YES b) ... c) 56*1, 28*2, 14*4, 8*7 d) + e) + then + + f) 2m(m _ _)(m_ _) g) 2m(m + 8)(m + 7) |
6k2p2
- 13kp + 6
...
NO 6*1, 3*2 6*1, 3*2 + - then - - ( __kp____ )( _kp____ ) (3kp - 2)(2kp - 3) |
z4 + 7z2
- 30
...
YES ...
- + then larger + (z2 _ __)(z2 _ _) (z2 + 10)(z2 - 3) |
18a2
- 3ab - 28b2
...
NO 18*1, 9*2, 6*3 28*1, 14*2, 7*4 - - then larger - (_ a _ _b)(_a _ _b) (6a + 7b)(3a - 4b) |
SPECIAL POLYNOMIALS
1. If the factors of the first term of a trinomial contain a perfect square, the factors of the last term contain a perfect square, and the signs that will be used are the same, try the perfect square factors first. It may be a perfect square-trinomial.
16k2 - 24k + 9 16 = 16*1, 8*2, 4*4 9 = 9*1, 3*3 16k2 - 24k + 9 = (4k - 3)(4k - 3) or (4k - 3)2
2. If no factors work to give the original polynomial this is referred
to as prime or non-factorable.
x2 - 4x + 53. A polynomial of the form x2 - y2 where the first term is a perfect square, the last term is a perfect square, and there is a minus sign in between is called the difference of two squares. Since the middle term is missing (0 + I), the 0 I terms must have been the same except for the signs. Factor by taking the square root of the first term and the square root of the last term. Place these in the appropriate positions in the parentheses with a plus in one and a minus in the other.
4x2 - 25y2 = (2x + 5y)(2x - 5y)4. A polynomial of the form x2 + y2 (sum of two squares) can only be factored if there is a GCF that can be taken out.
36x2 + 1 Prime
4x2 + 36y2 = 4(x2 + 9y2)