Integration Unit 8
Factoring and solving quadratics by Factoring
Name __________________________________
Period ________
Objective 1: Factors and Greatest Common Factors
Factors of a number: Numbers that divide evenly into a number are called factors.
They are the “sides” of the answer rectangle’s area.
This rectangle shows that
that
9 and 4 are factors of 36
36 is evenly divisible by 9
36 is evenly divisible by 4
This rectangle shows
8 and 12 are factors of 96
96 is evenly divisible by 8
96 is evenly divisible by 12
8 96 How to find Factors?
Use the multiplication table to find your “answer” in the large area: If you know the area of a figure (or
the answer), you can find the row and column that multiply together to create the answer.
Practice I: Find all the factor pairs for each number below.
a) 36 ____&____, ____&____, ____&____, ____&____, ____&____, ____&____,
____&____, ____&___, ____&____, ____&____, ____&____, ____&____, ____&____
b) 40 ____&____, ____&____, ____&____, ____&____, ____&____, ____&____,
____&____, ____&___, ____&____, ____&____, ____&____, ____&____, ____&____
If you know one of the factors, it makes it much easier:
Practice II: Find the missing factor:
a) _____ i 3 = 51
c) _____ i ( −12 ) = 96
b) _____ i 4 = 96
d) _____ i 13 = −182
Greatest Common Factor (GCF) of NUMBERS
The largest number that divides evenly into ALL numbers is called the GCF, the greatest common
factor.
Practice I: Name the GCF (if any) of each group of numbers below.
a) 30, 40, 55 GCF :________ b) 1, 8, 14 GCF :________ c) 12, 24, 30 GCF :_________
d) 12, 48, 66 GCF :________ e) 48, 64 GCF :_________ f) 27, -54, 108 GCF :________
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Greatest Common Factor (GCF) of letters (VARIABLES)
The letters can also divide evenly into ALL terms and be part of the GCF. Look for the most of any
letter that all the terms share, then divide it out.
Practice II
1) x 3 , x 4 , x 5 GCF :_________ 2) x, x 3 , x10 GCF :_________
3) x 3 y, x 4 y, xy 3 GCF :_________
4) x10 , x15 GCF :_________ 5) x 5 y 3 , x 3 y 4 , x 5 y 7 GCF :_________ 6) xyz, xy GCF :_________
The letters that divide evenly into ALL terms is called the GCF, the greatest common factor. Remember
that exponents subtract when dividing and negative exponents are NOT allowed!
Greatest Common Factor (GCF) of a polynomial: NUMBERS and VARIABLES
The greatest common factor (GCF) of a polynomial is the largest monomial that is a factor of (divides
evenly out of) each term of the polynomial. It can consist of a number, a letter, letters, or any
combination of numbers and letters.
Note: The GCF must be a factor of EVERY term in the polynomial.
•
•
FIRST: Find the largest number (if any) that evenly divides into EVERY coefficient and
constant in the terms of the polynomial.
THEN: Find the maximum amount of letter or letters that divide into EVERY term. Use the
lowest exponent on any letter.
Practice III: Name the GCF (if any) of each group of numbers below:
1) 15x 2 ,25x 3 ,10x 5 GCF :________
2) 8x,12x,20x 3 GCF :________
3) 15x 2 ,25x 3 ,10 GCF :_________
4) x 7 ,10x, 42x 3 GCF:________
5) 5x, 3,8y GCF:________
6) 14x 2 ,27x 3 , 42x 3 GCF:_______
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Objective 2: Factoring Polynomials with a GCF
Objective 2a: Identifying Factors of a Polynomial: Factors of a polynomial are monomials, binomials, trinomials, etc. that multiply with each other to create the original simplified polynomial. Getting Ready: Write the factored and simplified form of each rectangle shown below:
a) Factored form (side) x (side): _____________________ b) Factored form: ___________________
Simplified form (AREA): ____________________
Simplified form: ____________________
6x 18y
y 12 Factoring a polynomial (or “How to find the factors of a Polynomial”): Factoring a polynomial means to find the lengths of the “sides” of the simplified form area. How to Factor Polynomials with a GCF: Identify the GCF of the given polynomial. One side is the GCF, then figure out the other factor by
dividing.
Rewrite the original problem in “factored form”. Side * side.
Example 1: Factor 3x 2 + 6x Identify the GCF (one of the sides) ____________
Use the division or area model to find the “other (side) factor”.
Division Model
Divide to find the result
Area Model
Draw rectangle with GCF as one side. Find missing side.
3x 3x2 6x Ex 1 solution:
Factored form is __________________________________
Simplified form is _________________________________________
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Example 2: Factor 15x 3 + 20x 2 − 30x Identify the GCF (one of the sides____________
Use the division or area model to find the “other factor”.
Division Model
Divide to find the result
Area Model
Draw rectangle with GCF as one side. Find missing side.
5x 15x3 20x2 -­‐30x Ex 2 solution: Factored form is __________________________________
Simplified form is _________________________________________
Problem 1) Factor 4x 4 − 16x 3 + 8x 2 + 12x .
Problem 2) Factor 3x 4 y 3 + 12x 3 y − 18x 2 y 2 .
Identify the GCF _______________
Identify the GCF _______________
You have to be careful to look at all the numbers and variables. You
must be able to factor out of every term in order to identify the
GCF.
Use the division or area model to find the “other factor” of each problem.
Factored form: __________________________
Factored form: _____________________________
. Simplified form: _____________________ Simplified form _____________________________
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Practice I: Factoring polynomials with a GCF You are given the simplified form of the problem. Find its factors.
1) 8 x 2 + 16 x GCF:_________
2) 35 x 7 − 49 x 6 + 14 x 5 − 63 x 3
factored form: _____________________
factored form: _____________________
3) 2 x 3 + 6 x 2 − 10 x
GCF: _________
GCF: _________
Leading Coefficient in factored form must be positive: divide by -3
factored form: _____________________
5) 2 x 3 + 10 x 2 + 8 x
4) −3x − 6
GCF: _________
factored form: _____________________
6) −x 2 + 3x
GCF: _________
factored form: _____________________
GCF: _________
factored form: _____________________
7) −3y 6 − 5y 3 + 2y 2 GCF: _________
8) 9 x 4 − 15 x 3 + 3 x 2
factored form: _____________________
factored form: _____________________
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GCF: _________
How to Factoring polynomials with a GCF when the GCF is a binomial Sometimes one side of the rectangle has the GCF written as a “group” instead of separated.
Example: Factor x(x − 2) + 3(x − 2)
See the GCF? It is
Division model result:
x ( x − 2 ) 3(x − 2)
+
(x − 2)
(x − 2)
= x+3
Factored form:
( x − 2)
Area Model (harder!)
x 3 ( x + 2 )( x + 3)
Find simplified form: x(x − 2) + 3(x − 2) = _____________________________________________
Practice: Factor the following
1) x(2x + 5) + 2(2x + 5) GCF___________
2) x(3x − 1) − 5(3x − 1) GCF ___________
factored form:________________________
factored form: ________________________
Simplified form: ____________________________ Simplified form: ___________________________
3) 2x(x + 4) − 3(x + 4)
4) x(2x + 3) + (2x + 3)
GCF___________
factored form:________________________
GCF__________
factored form: ________________________
Simplified form: ____________________________ Simplified form: ___________________________
5) x 2 (3x − 1) + (3x − 1) GCF___________
6) 6x(x − 4) − (x − 4) GCF___________
factored form:________________________
factored form: ________________________
Simplified form: ____________________________ Simplified form: ___________________________
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Grouping to Factor
Polynomials can be put into pairs (grouped) and the common binomial is divided out each pair.
REMEMBER: EXAMPLE: Factor: 15x 3 + 5x 2 + 3x + 1
Step 1: Group the first and second term in ( ). Then group the third and fourth term in ( ). CHANGE the sign of each term in the second group if there is a − between the two groups. 15x 3 + 5x 2 + ( 3x + 1)
These are each row in the area model above
Step 2: Divide out GCF Monomial from each group: 5x 2 ( 3x + 1) + 1( 3x + 1) See area model above (
•
•
)
Factor out the GCF from the first two terms and from last two terms. KEEP the + or − sign between the two groups!! Step 3: Divide out GCF Binomial: ( 3x + 1) • 5x 2 + 1 or ( 3x + 1) 5x 2 + 1 (
)
(
)
• Factor out the GCF that is the common binomial factor in each group. It is in one set of ( ) • Write the remaining divided result in another set of ( ) Make sure there is a multiplication symbol i between the two sets of ( ) or NO symbol. You can check your answer by multiplying the factors to find if they equal your original polynomial Objective 3 Practice: Factor each completely using either model above. Answers on next page
3
2
1) 12x − 9x + 4x − 3
3
2
2) 2x + 5x + 6x + 15
3) 3x 3 − 4x 2 + 9x − 12
4) 12x 3 + 4x 2 + 3x + 1
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5) x 3 − x 2 − 2x + 2
6) 5x 3 − 10x 2 + 3x − 6
7) x 2 − 3x − 2x + 6
8) 35xy−5x−56y+8
9) 12xy−28x−15y+35
Scrambled answers
2
(4x +1)(3x+1)
2
(5x +3)(x−2)
(5x − 8)(7y − 1)
(x−2)(x+3)
(4x − 5)(3y − 7)
2
(x −2)(x−1)
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2
(3x +1)(4x−3)
2
(x +3)(3x−4)
2
(x +3)(2x+5)
Factor the following by Grouping: Find your answer in the set below and cross out the box above it. 1) x 2 + 3x + xk + 3k 2) a 2 − 2a + ad − 2d 3) uv 2 + 5u + v 2 + 5v 4) x 2 − xk + 4x + −4k 5) ad + 3a − d 2 − 3d 6) y 3 + y 2 + 2y + 2 7) m 3 + m 2 n + 2mn 2 + n 3 8) u 3 − u 2 v + uv 2 − v 3 9) t 2 + 2t + 3kt + 6k 10) 2ab + 14a + b + 7 11) m 2 + mn − 3m − 3n 12) 5x 2 y − x 2 + 5y − 1 Put the remaining letters together to answer the following: How did Snidely Spellbinder Write a 4-­‐letter word That Begins and Ends with “E”? Page 11
Factoring Trinomials by Grouping: Example: Factor 6x 2 + 7x + 2 Find ac : a i c = 6 i 2 = 12 and find b. b = 7 + * Rewrite your trinomial as a 4-­‐term polynomial by replacing Adds to 7: 12x with and . 3x and 3x+4x 4x will become OR (The order doesn’t matter) OR Factor using grouping: EX 1) 5x 2 − 4x − 1 ac = _____ b = ____ EX 2) 4x 2 − 11x + 7 ac = _____ b = ____ Factor each trinomial below using Grouping Factor the following trinomials by grouping. Once you have both factors, find them in the answer box on the next page and cross out each factor. 1) 6x 2 + 19x + 3 ac = _____ b = ____ 2) 5x 2 − 9x − 2 ac = _____ b = ____ 3) 9x 2 + 15x + 4 4) 7x 2 + x − 8 ac = _____ b = ____ ac = _____ b = ____ Page 12
5) 2x 2 − 21x + 40 ac = _____ b = ____ 7) 8m 2 − 5m − 3 ac = _____ b = ____ 9) 14m 2 + 17m − 22 ac = _____ b = ____ Page 13
6) 15m 2 + 19m + 6 ac = _____ b = ____ 8) 4m 2 − 17m + 18 ac = _____ b = ____ 10) 3n 2 − m − 30 ac = _____ b = ____
Quick Factoring! (optional)
Factoring when a=1
Factoring ax 2 + bx + c when a = 1 means there will be no GCF to take out of the groups, other than x.
2
Example 1: Factor x + 8x + 12
6
x
x2
6x
2
2x
12
p
−4
p
p2
−4 p
−5
−5 p
20
ac = 12 , b = 8 Factored form: (x + 6)(x + 2)
Example 2: Factor p 2 − 9 p + 20 .
x
ac = 20 , b = −9 Factored form: ( p − 4)( p − 5)
Practice I: Factor the following
1)
n2 + 3n – 18
ac = _____ b = ____
3) r2 + 4r – 12
ac = _____ b = ____
5) x2 – 8x + 15
ac = _____ b = ____
2) y2 – 5y + 6 ac = _____ b = ____
4) n2 – n – 6
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ac = _____ b = ____
Special Cases
The following polynomials have specific patterns that make it easy to factor them without having to
make them into a 4-term polynomial (although you can.
Difference of Squares: a 2 − b 2 = (a + b)(a − b)
a
b
a
a2
ab
−b
−ab
−b 2
a
b
a
a2
ab
b
ab
−b 2
Notice the original polynomial has NO GCF and NO middle term. The middle term CANCELLED Perfect Square Trinomials:
a 2 + 2ab + b 2 = (a + b)(a + b) = ( a + b )
2
a 2 − 2ab + b 2 = (a − b)(a − b) = ( a − b )
2
a
−b
a
a2
−ab
−b
−ab
−b 2
The middle term DOUBLED Practice II: Factor each polynomial completely. Try to factor without grouping if you can.
1) 16x2 −9
2) 4x2 −25
3) 16x2 −40x+25
4) x2 −1
5) 9x2 −1
6) x2 −25
9) x2+2x+1
10) 49x2 −100
12) 49x2 −56x+16
7) x2 −100
8) x2 −9
11) 100x2 + 180x + 81
13) 4x2 −4x +1
14) x2 −49
15) x2 +10 x +25
Answers (scrambled)
(3x + 1)(3x − 1)
(x + 5)(x − 5)
(2x + 5)(2x − 5)
(x +10)(x −10)
(x+3)(x−3)
(4x − 5)
(2x − 1)
2
(10x + 9)
2
2
(x +7)(x −7)
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(7x + 10)(7x − 10)
(4x + 3)(4x − 3)
(7n + 10)(7n − 10)
(x + 5)
2
(7n − 4)
2
Using all methods: Factor the following using the various methods of factoring HINTS: #1-­‐5, 7, 10, 11, 14 and 16 are shortcuts and special cases. All other problems require creating 4-­‐term polynomial (grouping). Show your work on the next page Page 16
Work page for page 16 #6 work #9 work #13 work #17 work #8 work #12 work #15 work #18 work Page 17
Using Factoring to Solve Quadratics: The Zero-product property
When two numbers multiply to zero, one of the numbers must be 0. Example: 4 * 0 = 0 0 * -­‐3 = 0 When two factors multiply to zero, one of the factors must equal 0. We can factor a trinomial to SOLVE it using the zero product property. (what other method could we have used to solve this problem) Practice: Solving with Zero-­‐product Property Solve the following factored polynomials using the zero product property 1) x(x – 9) = 0
2) (9x – 2)(5x + 1) = 0
3) 5x(2x + 7) = 0
4) 7x(3x – 1) = 0
5) (2x – 1)(x + 4) = 0
6) (3x + 2)(3x – 2) = 0
7) (2x + 2)(7x + 6) = 0
8) 4x(x + 11) = 0
9) x(4x + 7) = 0
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Solving by FACTORING
Factor the problems below completely, then use the zero product property to solve.
HINTS:
Problems #1-5, and #14 can be factored using Shortcuts and do not require making a 4-term
polynomial (grouping)
Problem #6 and #12 have a GCF and should not be grouped. Just factor out the GCF and solve.
Problems #7-11 and #13 require making a 4-term polynomial in order to factor.
1) a 2 + 7a + 10 = 0 ac = _____ b = ____
3) y 2 − 49 = 0
5) u 2 − 7a − 18 = 0 ac = _____ b = ____
2) n 2 − 8n + 12 = 0
7) 2t 2 + 5a − 3 = 0
ac = _____ b = ____
8) 3w 2 − 8w + 4 = 0 ac = _____ b = ____
9) 2x 2 − 3x − 5 = 0
ac = _____ b = ____
10) 5v 2 + 29v + 20 = 0
ac = _____ b = ____ 4) x 2 + 5a − 6 = 0 ac = _____ b = ____ 6) m 2 − 5m = 0 GCF: ___________
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ac = _____ b = ____
11) 6n 2 − 19n + 15 = 0
13) 3b 2 + b − 10 = 0
12) 2k 2 + 7k = 0 GCF: ___________
ac = _____ b = ____
ac = _____ b = ____
14) 4y 2 − 25 = 0
What is the Title of this picture?
Find your answers and put them on the blank corresponding to the problem
number.
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Using the zero product property to solve quadratic equations
On the next sheet of paper:
• Write down the problem
• Factor the polynomial (grouping required for #7-11, 13.
• Use the zero product property to solve the equation. Check your answers on the puzzle.
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