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STATION 1: FACTORING NUMBERS AND MONOMIALS
Factors are numbers you multiply together to get a product. Because 3 x 4 = 12 we say 3 and 4 are
factors of 12. Prime numbers are numbers that have exactly 2 factors 1 and the number itself (like 5).
Composite numbers are numbers that have more than 2 factors (like 12).
List the factors of each number and classify the number as prime or composite.
1.
60
4.
23
2.
11
3.
5.
50
64
To completely factor monomials they are written as the product of their prime factors. To write 45 in
factored form 45 = 3 * 3 *5. To list 12x2y in factored form – 2 * 2* 3* x * x * y.
Factor each monomial.
1.
4.
8x2
28a3b4c
2. -4m2n
3.
-16x3y4
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STATION 2: FINDING THE GCF OF MONOMIALS AND POLYNOMIALS
The Greatest Common Factor (GCF) of a set on monomials or a polynomial is the largest factor that the
terms of a set of monomials or a polynomial have in common. To find the GCF you need to factor the
terms of a set of monomials or a polynomial and determine all prime factors they have in common,
then multiply those prime factors together.
For example, find the greatest common factor between 15ab2 & 12a2b2
15ab2 =
12a2b2 =
3*5*a*b*b
2*2*3*a*a*b*c
GCF = 3 * a * b * b = 3ab2
Find the greatest common factor (GCF) of each set of numbers or monomials.
1. 7ax, 21ax2, 35ax3
3. 20, 24
2.
6p, 3p2, 33px
4. 48x2y3z, 54xy2z, 36x3y3x3
The first step in factoring polynomials is to factor out the GCF. First identify the GCF of a polynomial,
then divide EACH term by that GCF, list your answer as GCF(what is left after dividing polynomial by
GCF).
Factor the polynomial 15ab2 + 12a2b2. We already know that the GCF of the terms in the polynomial is
3ab2, so we will divide each term by 3ab2.
15ab2 + 12a2b2 = 3ab2(5 + 4a)
Remember you rules of powers – when dividing a powers SUBTRACT exponents. If you have a
negative exponent – FLIP IT!!
You can check your answer by using the distributive property and seeing if you get your original
polynomial.
Factor each polynomial. If the polynomial cannot be factored, write prime.
1. 4x2 – 8x + 12
2. 6x3 – 15x2
4. 7x2 -21x
5. 9x6 + 81x3 – 27x
3. 7ab + 13c
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STATION 3: DIVIDING A POLYNOMIAL BY A MONOMIAL
When asked to find a quotient you divide the given polynomial by the monomial and just list what is
left after you have successfully divided the polynomial. Remember you rules of powers – when
dividing powers SUBTRACT exponents. If you have a negative exponent – FLIP IT!!
For example: Find the quotient:
(18x3 + 48x2 -24x) ÷ 6x = (3x2 + 8x – 4) Remember, you can only combine LIKE terms
Like terms with variables mean the variables are the same
And the exponents are the same.
LIKE TERMS: 4x2 & 7x2
NOT LIKE TERMS: 4x2 & 7x3
Find each quotient.
1. (15x – 10y2) ÷ 5
2. (6ac + 2bc) ÷ 2c
3. (8g2j – 12g3hj) ÷ 4g
4. (30cd2 -10bc3d) ÷ 5c
5. (30f2g2h + 21gh2) ÷ 3gh
6. (64z – 80z3 – 16z2) ÷ 8z
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STATION 4: FACTORING TRINOMIALS IN THE FORM: x2 + bx + c
When factoring a polynomial of the form x2 + bx + c (where the coefficient of your first term is 1) you
focus on the factors of the last term, determine the sign of your binomial factors, and then determine
which set of factors when combined will give you your middle term.
Factor:
x2 + 5x + 6
(x + )(x + )
(x +2)(x + 3)
The factors of 6 are 1, 6, 2, 3
The sign of the last term is positive, the only way to get a positive
Product is to multiply (+, +) or (-, -)
The middle term of the polynomial is positive, so our binomial signs (+, +)
Find the factors of 6 that when added together give you 5
Remember, before finding the binomial factors of a polynomial check to see if the GCF has been
factored out. For example before trying to find the binomial factors of 4x2 + 24x + 36 make your life
easier by factoring out a 4!!!
You can always (and you should always) check your answer by using the foil method to see if when
you multiply your factors together you get the original polynomial!
Factor each trinomial. If the trinomial cannot be factored, write prime. Circle your final
answer. Show me how you checked your work!
1. s2 + 7s + 10
2. x2 – x – 6
3. 5y2 – 20y -25
4. b2 – b + 8
5. 3a2 +9a +6
6. 3c2 + 12c + 15
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STATION 5: FACTORING TRINOMIALS IN THE FORM: ax2 + bx + c
When factoring polynomials of the form ax2 + bx +c follow these steps:
10x2 + 8x – 2
Check to see if you can factor anything out of all 3 terms.
You can factor out a 2
2(5x2 + 8x -1)
2(
-
)(
+
)
Now list the factors of your
first term 5x2: 1x,5x
and your last term 1: 1,1
Since my last term is negative, one factor will have a negative
and the other factor will have a positive.
2(5x + 1)(x - 1)
Plug in factors of the first and last terms – guess and check
Does this give your middle term? Multiply outer and inner terms & combine
Outer: 5x * -1 = 5x Inner: 1 * x = x Combine: -5x + 1x = -4x
This is close to our answer but the sign is wrong
Switch the signs in your factors
2(5x – 1)(x + 1)
Outer: 5x * 1 = 4x Inner: -1 * x = -1x Combine: 5x -1x = 4x CORRECT!
Keep trying to get the correct answer
You can and should check your final answer by multiplying out and seeing if
you get the original polynomial.
Factor each trinomial. If the trinomial cannot be factored, write prime. Circle your final
answer. Show me how you checked your work!
1. 3x2 + 7x – 6
2. 9x2 – 6x -3
3. 5x2 + 9x – 2
4. 6x2 + 10x +4
5. 3x2 -7x -6
6. 20x2 -33x +10
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STATION 6: FACTORING TRINOMIALS, SPECIAL FACTORS
When factoring trinomials if your first and second term are perfect squares (meaning that you
can take the square root of the number and the square root is a whole number) then you should
try the square roots as factors.
There are two special cases
1. Perfect Square Trinomials:
a2 +2ab +b = (a +b)2
a2 -2ab +b = (a – b)2
x2 + 10x +25 = (x + 5)2
x2 – 10x +25 = (x – 5)2
2. Different of Two Squares: remember difference means subtraction!
a2 – b2 = (a – b)(a + b)
36x2 – 49y2
Factor each trinomial. If the trinomial cannot be factored, write prime. Circle your final
answer. Show me how you checked your work!
1. 121x2 – 64y2
2. g2 + 8g +16
3. 4k2 + 20k + 25
4. 25z2 -16
5. 49y2 + 25
6. 18x2 +12x +2
7. 4x2 -24x +36
8. 16e2 - 4