Factoring and Box Method
Transcript of Factoring and Box Method
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Algebra 2
Factoring Basics
&
Box Method
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Factoring Polynomials
This process is basically the REVERSEof the distributive property.
)5)(2( xx 1032 xx
distributive property
factoring
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1032 xx
In factoring you start with a polynomial (2 or more terms) and you want to rewrite it as a product (or as a single term)
Factoring Polynomials
Three terms
)5)(2( xx
One term
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Techniques of Factoring Polynomials
1. Greatest Common Factor (GCF). The GCF for a polynomial is the largest monomial that divides each term of the polynomial.
Factor out the GCF: 23 24 yy
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Factoring Polynomials - GCF
23 24 yy
y2
yyy22
Write the two terms in the form of prime factors…
They have in common 2yy
)12(2 2 yy
yy2
1)(2yy
This process is basically the reverse of the distributive property.
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Check the work….
)12(2 2 yy 34y 22y
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Factoring Polynomials - GCF
Factor the GCF:
24233 8124 cabcbaab
3 terms
4ab2( )b - 3a c2 + 2b c2 2
One term
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Factoring Polynomials - GCF
)(
EXAMPLE:
)42(3)42(5 xxx
)42( x 5x - 3
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Examples
Factor the following polynomial.
)53(4
)53(4
54432012
22
42
xx
xxxx
xxxxxxxx
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Examples
Factor the following polynomial.
)15(3
)15(3
353315
42
42
42534253
xyyx
yxyx
yxyxyxyx
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Techniques of Factoring Polynomials
2. Factoring a Polynomial with four or more Terms by Grouping
)2()3(
)3(2)3(
623
2
2
23
xx
xxx
xxx There is no GCF for allfour terms.
In this problem we factor GCFby grouping the first two terms and the last two terms.
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To be continued….
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3. Factoring Trinomials.
652 xx We need to find factors of 6
Since 6 can be written as the product of 2 and 3and 2 + 3 = 5, we can use the numbers 2 and 3 to factor the trinomial.
….that add up to 5
Techniques of Factoring Polynomials
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Factoring Trinomials, continued...
652 xx 2 x 3 = 62 + 3 = 5
Use the numbers 2 and 3 to factor the trinomial…
Write the parenthesis, with An “x” in front of each.
3)2( xxWrite in the two numbers we found above.
xx )(
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652 xxYou can check your work by multiplying back to get the original answer
3)2( xx
3)2( xx
6232 xxx
652 xx
So we factored the trinomial…
Factoring Trinomials, continued...
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Factoring Trinomials
61
65
67
2
2
2
xx
xx
xx
Find factors of – 6 that add up to –5
Find factors of 6 that add up to 7
Find factors of – 6 that add up to 1
6 and 1
– 6 and 1
3 and –2
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61
65
67
2
2
2
xx
xx
xx
factors of 6 that add up to 7: 6 and 1
1)6( xx
factors of – 6 that add up to – 5: – 6 and 1
factors of – 6 that add up to 1: 3 and – 2
1)6( xx
2)3( xx
Factoring Trinomials
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Factoring TrinomialsThe hard case – “Box Method”
62 2 xx
Note: The coefficient of x2 is different from 1. In this case it is 2
62 2 xx
First: Multiply 2 and –6: 2 (– 6) = – 12
1
Next: Find factors of – 12 that add up to 1– 3 and 4
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Factoring TrinomialsThe hard case – “Box Method”
62 2 xx
1. Draw a 2 by 2 grid.2. Write the first term in the upper left-hand corner 3. Write the last term in the lower right-hand corner.
22x6
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Factoring TrinomialsThe hard case – “Box Method”
62 2 xx – 3 x 4 = – 12– 3 + 4 = 1
1. Take the two numbers –3 and 4, and put them, completewith signs and variables, in the diagonal corners, like this:
22x
6
It does not matter whichway you do the diagonal entries!
Find factors of – 12 that add up to 1
–3 x
4x
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The hard case – “Box Method”1. Then factor like this:
22x6x3
x4
Factor Top Row Factor Bottom Row
2
22x6x3
x4x
From Left Column From Right Column
22x6x3
x42x
x222x
6x3
x4
x2
2x
3
x
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The hard case – “Box Method”
22x6x3
x4
x2
2x
3
)32)(2(62 2 xxxx
Note: The signs for the bottom rowentry and the right column entry come from the closest term that youare factoring from. DO NOT FORGET THE SIGNS!!
++
Now that we have factored our box we can read offour answer:
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The hard case – “Box Method”
24x12
x16x3
x
3
x44
12194 2 xx
Finally, you can check your work by multiplying back to get the original answer.
Look for factors of 48 that add up to –19 – 16 and – 3
)4)(34(12194 2 xxxx
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Use “Box” method to factor the following trinomials.
1. 2x2 + 7x + 3
2. 4x2 – 8x – 21
3. 2x2 – x – 6
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Check your answers.
1. 2x2 + 7x + 3 = (2x + 1)(x + 3)
2. 2x2 – x – 6 = (2x + 3)(x – 2)
3. 4x2 – 8x – 21 = (2x – 7)(2x + 3)
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Note…
Not every quadratic expression can befactored into two factors.
• For example x2 – 7x + 13.
We may easily see that there are no factors of 13 that added up give us –7
• x2 – 7x + 13 is a prime trinomial.
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Factoring the Difference of Two Squares
The difference of two bases being squared, factors as the product of the sum and difference of the bases that are being squared.
a2 – b2 = (a + b)(a – b) FORMULA:
(a + b)(a – b) = a2– ab + ab – b2 = a2 – b2
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Factoring the difference of two squares
Factor x2 – 4y2 Factor 16r2 – 25
(x)2 (2y)2
(x – 2y)(x + 2y)
Now you can check the results…
(4r)2 (5)2
Difference of two squares
DifferenceOf two squares
(4r – 5)(4r + 5)
a2 – b2 = (a + b)(a – b)
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Difference of two squares
)4)(4(
)4()(
16
22
2
yy
y
y
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Difference of two squares
)95)(95(
)9()5(
8125
22
2
xx
x
x
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Difference of two squares
)4)(2)(2(
)4)(4(
)4()(
16
2
22
222
4
yyy
yy
y
y