1. Warm up
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2. Factoring Polynomials
Digital Lesson
Factoring Polynomials

3. Factoring these trinomials is based on reversing the FOIL process.
To factor a simple trinomial of the form x2 + bx + c, express the trinomial as the product of two binomials. For example,
x2 + 10x + 24 = (x + 4)(x + 6).
Factoring these trinomials is based on reversing the FOIL process.
Example: Factor x2 + 3x + 2.
Express the trinomial as a product of two binomials with leading term x and unknown constant terms a and b.
x2 + 3x + 2
= (x + a)(x + b) F O I L
= x2
+ bx
+ ax
+ ba
Apply FOIL to multiply the binomials.
= x2 + (b + a) x + ba
Since ab = 2 and a + b = 3, it follows that a = 1 and b = 2.
= x2 + (1 + 2) x + 1 · 2
(Product-sum method)
Therefore, x2 + 3x + 2 = (x + 1)(x + 2).
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Factor x2 + bx + c

4. It follows that both a and b are negative.
Example: Factor x2 – 8x + 15.
x2 – 8x + 15
= (x + a)(x + b)
= x2 + (a + b)x + ab
Therefore a + b = -8
and ab = 15.
It follows that both a and b are negative.
Sum
Negative Factors of 15
- 1, - 15
-15
-3, - 5
- 8
x2 – 8x + 15 = (x – 3)(x – 5).
Check:
= x2 – 8x + 15.
(x – 3)(x – 5) = x2 – 5x – 3x + 15
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Example: Factor

5. two positive factors of 3613 36
Example: Factor x2 + 13x + 36.
x2 + 13x + 36
= (x + a)(x + b)
= x2 + (a + b) x + ab
Therefore a and b are:
two positive factors of 36
Sum
Positive Factors of 36
whose sum is 13.
1, 36 37
2, 18 20 15
3, 12
4, 9 13
6, 6 12
= (x + 4)(x + 9)
x2 + 13x + 36
Check: (x + 4)(x + 9)
= x2 + 9x + 4x + 36
= x2 + 13x + 36.
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Example: Factor

6. Example: Factor Completely
A polynomial is factored completely when it is written as a product of factors that can not be factored further.
Example: Factor 4x3 – 40x x.
4x3 – 40x x
The GCF is 4x.
= 4x(x2 – 10x + 25)
Use distributive property to factor out the GCF.
= 4x(x – 5)(x – 5)
Factor the trinomial.
Check:
4x(x – 5)(x – 5)
= 4x(x2 – 5x – 5x + 25)
= 4x(x2 – 10x + 25)
= 4x3 – 40x x
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Example: Factor Completely

7. You try… Factor x2 + 7x + 12 Factor x2 – 12x + 27
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8. Solve equations by factoring
Solve x2 + 2x = 15.
x2 + 2x = 15 original equation
x2 + 2x – 15 = 0 rewrite the equation so that one side equals 0
(x – 3)(x + 5) = 0 factor
x – 3 = 0 or x + 5 = 0 zero product property
x = 3 x = -5
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9. Real world problem by factoring
Marion has a small art studio in her back yard. She wants to build a new studio that has three times the area of the old studio by increasing the length and width by the same amount. What are the new dimensions of the new studio?
Existing studio
12 ft
10 ft x
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10. x2 + 22x – 240 = 0 rewrite the equation so that one side equals 0
Current area: 12 x 10 = 120
New area: 3(120) = 360
(x + 12)(x + 10) = 360
x2 + 22x = 360 foil
x2 + 22x – 240 = 0 rewrite the equation so that one side equals 0
(x + 30)(x – 8) = 0 factor
x + 30 = 0 x – 8 = 0
x = x = 8
Existing studio
12 ft
10 ft x
length
width
Since we are talking about length, we know we can not have negative distance therefore our answer is x = 8
New dimensions length: = 20 ft
width: = 18 ft
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11. Page 493 # 18 – 51 multiples of 3  Class work…
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