Download 9-2 Factoring Using the Distributive Property - lindsey-math

9-1 Factoring and Greatest Common Factors
**Prime Number: Only has 2 factors: 1 and itself.
(Those are the only 2 numbers you can multiply together
to get that number.)
EXAMPLES: 2 (b/c 2 x 1 = 2) 7 (b/c 7 x 1 = 7)
3 (b/c 3 x 1 = 3) 11 (b/c 11 x 1 = 11)
5 (b/c 5 x 1 = 5) 13 (b/c 13 x 1 = 13)
**Composite Number: Has MORE than 2 factors (There
is more than 1 way to multiply numbers to get that
number)
EXAMPLES: 4 (b/c 4 x 1 = 4 AND 2 x 2 = 4)
6 (b/c 6 x 1 = 6 AND 2 x 3 = 6)
12 (b/c 12 x 1 = 12 AND 2 x 6 = 12 AND 4 x 3 = 12)
**Prime Factorization: When you take a number and
find all the PRIME numbers that you can multiply to get
that number.
**A good way to find the prime factorization is to do a
FACTOR TREE: (break down each number with “what
times what equals that number” until you get ALL prime
numbers)
EXAMPLE: Find the prime factorization of 90
90
^
**When you reach ALL Prime
9 x 10
numbers, ALL the prime
^
^
numbers are your answer….
3x3 2x5
So… 2 x 3 x 3 x 5 is your answer!
**When you factor a negative number:
EXAMPLE: Find the prime factorization of -90.
STEP 1: Always factor -1 and the number
first…then factor like normal.
90
^
** So… -1 x 2 x 3 x 3 x 5 is
-1 x 90
your answer b/c they are all
^
prime numbers and they
9 x 10
equal -90 if you multiply
^
^
them.
3x3 2x5
**When you FACTOR VARIABLES (monomials), do the
numbers the SAME way (factor tree)…then the
Variables, just LIST THEM ALL!!!
EXAMPLE: Factor 12 a²b³
12
^
**Answer: 2 · 2 · 3 · a · a · b · b · b
2x6
b/c 2 · 2 · 3 = 12 and there are 2 “a’s”
^
and 3 “b’s”
2x3
**Greatest Common Factor (GCF): “Ask yourself
“What’s the BIGGEST number that goes into BOTH
numbers…”
**If there are VARIABLES, Ask yourself, “What’s the
MOST amount of each letter that goes into both?”
EXAMPLE: Find the GCF of 10x²y and 15x^4y³z
**ANSWER: 5x²y
(B/c 5 is the biggest number
to go into 10 and 15. Only x² will go into BOTH… and
only y will go into BOTH, and z does not go into either!
**If the GCF is 1…it is called “Relatively Prime.”
9-2 Factoring Using the Distributive Property
**Factoring a Polynomial:
1. Figure out the GCF (or biggest number that goes
into BOTH numbers…..AND the MOST variables that go
into BOTH.
2. Write that number down (OUTSIDE the
parenthesis)
3. Divide that number by each monomial—(That’s
what goes INSIDE the parenthesis)
EXAMPLE 1: Factor 12a² + 16a
STEP 1: Figure out the GCF…“4a” goes into both
STEP 2: Divide 4a into BOTH 12a² and 16a
12a² + 16a = 3a + 4
4a
4a
SO… 4a(3a + 4)
**Factoring by Grouping:
*Group if there are 4 or more terms
*Group if common factors can be grouped together
EXAMPLE 2: 4ab + 8b + 3a + 6
STEP 1: Group terms with common factors
(4ab + 8b) + (3a + 6)
STEP 2: Factor out the GCF in EACH group (what
number and letter goes into BOTH numbers…put that
answer outside parenthesis….then divide by it…that
number in parenthesis)
4b(a + 2) + 3(a + 2)
(4b went into both 4ab and 8b) and (3 went into both 3a
and 6)
STEP 3: If what is in parenthesis is the SAME, then
you are correct so far (both are (a + 2). What is IN
parenthesis is one answer. What is OUTSIDE
parenthesis is the other answer.
(4b + 3)(a + 2)
**When a Polynomial EQUALS ZERO, you can find an
actual answer for x. Set BOTH answers in parenthesis
equal to zero. (make 2 different problems.)
EXAMPLE 3: (x – 5)(3x + 4)
STEP 1: Put (x – 5) = 0 AND (3x + 4) = 0 and solve
each one for x
x–5=0
3x + 4 = 0
+ 5 +5
+4 +4
x =5
3x
=4
3
3
So….x = 5 AND x = 4/3
EXAMPLE 4: x² = 7x
STEP 1: IF there is an EQUAL sign, you have to rearrange it so that the polynomial EQUALS ZERO
x² = 7x (subtract 7x to both sides)
-7x -7x
x² -7x = 0
STEP 2: if you can FACTOR something out, then
factor first!
x(x-7) = 0
STEP 3: set them both equal to 0 and solve
x=0
x-7=0
+7 +7
x=7
9-3 Factoring Trinomials
Step 1: Multiply the 1st term and the 3rd term together
Step 2: Find all the factors of that number
Step 3: Pick the factors that ADD up to make the middle
term
Step 4: RE-WRITE the 1st term, put in the 2 new terms,
then re-write the 3rd term.
Step 5: Put parenthesis around the first 2 terms and the
last 2 terms.
Step 6: Factor out what you can in EACH parenthesis
Step 7: What is IN parenthesis is ONE part of your
answer…..what is OUTSIDE parenthesis is the
OTHER part of your answer!
Step 8: You can “FOIL” it to see if you get your original
problem to CHECK yourself!
EXAMPLE: x² + 6x + 8
Step 1: Multiply x² and 8 and you get 8x².
Step 2: Find all the factors of 8x².
8x²
1x 8x
2x 4x
Step 3: Decide which ones add up to the middle term (6x)
2x + 4x = 6x … So…use 2x and 4x
Step 4: Re-write 1st term, 2 new terms and last term:
x² + 2x + 4x + 8
Step 5: Put parenthesis around first 2 terms and last 2
(x² + 2x) + (4x + 8)
Step 6: Factor what you can in each
x(x + 2) + 4 (x + 2)
Step 7: (x + 2) is IN parenthesis, so that is part of my
answer, and (x + 4) is OUTSIDE parenthesis, so that is
the OTHER part of my answer!! So….(x + 2)(x + 4)
Step 8: “FOIL” to check your answer:
x² + 4x + 2x + 8 =
x² + 6x + 8 YES, it is correct!
EXAMPLE: x² - 7x – 18
Step 1: Multiply -18 and x² and get -18x².
Step 2: Find all factors of -18x²
-18x 1x
18x -1x
-2x 9x
2x -9x
-6x 3x
6x -3x
Step 3: Pick the factors that ADD up to the middle term
(-7x).
2x + -9x = -7x
Step 4: Re-write: x² + 2x + -9x + -18
Step 5: Put parenthesis around first 2 and last 2
(x² + 2x) + (-9x + -18)
Step 6: Factor out what you can in each parenthesis
x(x + 2) + -9(x + 2)
Step 7: INSIDE Parenthesis (x + 2)…..OUTSIDE
parenthesis (x + -9) So…answer: (x + 2)(x + -9)
Step 8: Foil and check (x + 2)(x + -9)
x² - 9x + 2x – 18
x² -7x -18 YES ….CORRECT!!
More of 9-3: Factoring and Solving Trinomials
**If the Trinomial is set equal to ZERO….then we can
get actual answers for X!!!
EXAMPLE: x² + 7x + 6 = 0
Step 1: Factor the trinomial using the same 8 steps as
before: 6x²
6x 1x = 7x
(x² + 6x) + (1x + 6)
2x 3x
x(x + 6) 1(x + 6)
(x + 6)(x + 1)
Step 2: Set EACH parenthesis equal to 0 (make 2
separate equations) and solve for x
(x + 6) = 0
(x + 1) = 0
-6 -6
-1 -1
x = -6
x = -1
EXAMPLE: x² + 5x = 6
Step 1: In order to solve a trinomial….Make sure it is
EQUAL to ZERO and in the form: x² + bx + c!! In this
one, SUBTRACT 6 from both sides…and set it equal to 0.
x² + 5x
=6
-6 -6
x² + 5x – 6 = 0
Step 2: Factor the Trinomial using the 8 steps
-6x²
-6x 1x
(x² + 6x) + (-1x -6)
6x -1x = 5x
x(x + 6) -1(x + 6)
(x + 6)(x – 1)
Step 3: set each one equal to 0 and solve for x
(x + 6) = 0
(x – 1) = 0
-6
-6
+1 +1
x
= -6
x = 1
9-4 Factoring Trinomials (MORE!!)
**Use the SAME 8 Steps that we have been using! (The
only difference is there will be a number in front of the
x².)
EXAMPLE: 6x² + 17x + 5
Step 1: multiply 6x² × 5 and get 30x².
Step 2: Find the factors of 30x²
30x 1x
2x 15x
Step 3: 2x + 15x = 17x (the middle term)
Step 4: re-write: (first term, 2 new terms, last term)
6x² + 2x + 15x + 5
Step 5: Put parenthesis around the first 2 and last 2
(6x² + 2x) + (15x + 5)
Step 6: factor what you can
2x(x + 1) 5(x + 1)
Step 7: (x + 1) (2x + 5)
Step 8: FOIL to check: x² + 2x + 5x + 5
x² + 7x + 5
**If the Trinomial can NOT be factored, then it is called
a PRIME POLYNOMIAL.
EXAMPLE: 2x² + 5x – 2
Step 1: Multiply 2x² × -2 and get -4x².
Step 2: Find the factors of -4x².
-4x 1x
4x -1x
-2x 2x
Step 3: NONE of the factors add up to 5x…so it is a
PRIME POLYNOMIAL.
EXAMPLE: -16x² + 42x + 6 = 26
Step 1: You always need to make sure it’s in the form
“ax² + bx + c.” So in this example, subtract 26 from both
sides.
-16x² + 42x + 6 = 26
-26 -26
-16x² + 42x – 20 = 0
Step 2: 2 goes into all three terms, so factor -2 out.
-2(8x² - 21 + 10) = 0
Step 3: If you divide -2 to both sides, you can get rid of
the -2 totally to make 8x² - 21 + 10 = 0
Step 4: Factor 8x² - 21 + 10 = 0 using the 8 steps.
80x²
-16x -5x
(8x² - 16x) + (-5x + 10)
8x(x – 2) -5(x – 2)
(8x - 5)(x – 2) = 0
Step 3: set each one equal to 0 and solve
8x – 5 = 0
x-2=0
-5 -5
+2 +2
8x
= -5
x = 2
8
8
x = -5/8 and x = 2
9-5 Factoring Differences of Squares
To factor “Differences of Squares,”
EXAMPLE: x² - 25
Step 1: Know that one binomial is going to be positive,
and the other one is going to be negative.
Step 2: Take the square root of the first term, and the
square root of the second term….and make binomials—
one that is positive and one that is negative!)
(x – 5)(x + 5)
(b/c “x” is the square root of x², and 5 is the square root
of 25. I made one positive and the other negative!)
EXAMPLE: 49h² - 16
Step 1: square root of 49 is 7, and square root of 16 is 4:
(7h – 4)(7h + 4)
EXAMPLE: 48x³ - 12x = 0
Step 1: Sometimes you can factor out a GCF first (since
12x goes into both 48x³ and -12x, FACTOR 12x FIRST!)
12x(4x² - 1)
Step 2: Then put the 12x to the “side” for a bit….take the
square root of 4 and 1, and make binomials
(2x + 1)(2x – 1)
Step 3: NOW…put the 12x back into your answer…so
your answer should be: 12x(2x +1)(2x – 1) = 0
Step 4: Since there are EQUAL signs, you can SOLVE
for X!!! Set each one equal to zero and solve for x.
12x = 0
2x + 1 = 0
2x – 1 = 0
x=0
x = -1/2
x=½
9-6 Perfect Squares and Factoring
*Perfect Square: is something like:
(x – 2)²
or (x – 2)(x – 2)
OR
(x + 4)²
or (x + 4)(x + 4)
EXAMPLE: x² + 8x + 16
Step 1: Factor it the way you like to factor (get 2
binomials!)
(x + 4)(x + 4)
Step 2: Decide if it’s a perfect square! YES, it is because
each binomial is the same!