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Published byAugustine Joseph Modified over 9 years ago
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Factoring means finding the things you multiply together to get a given answer.
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You did some work with factoring in grade school. For instance, you found the prime factorization of numbers.
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You have also found factors of numbers and their common factors.
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When adding or subtracting fractions, you used factors to find the least common denominator.
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In algebra we mostly care about factoring polynomials. We want to find what you need to multiply together to get a given polynomial. It’s like you’re playing Jeopardy with the distributive property.
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Almost all the time we will be factoring quadratic trinomials.
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The most common factoring problems look like this: Factor x 2 + 12x + 35 We need to find the quantities we can multiply to get this polynomial.
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Factor x 2 + 12x + 35 The answer will have the format (x + ___)(x + ___)
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Factor x 2 + 12x + 35 To find the numbers that go in the quantity, find what you can multiply to get 35 that adds up to 12
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Factor x 2 + 12x + 35 multiply to get 35 that adds up to 12 The only numbers that do both are 7 and 5. So … (x + 7)(x + 5)
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Factor x 2 + 12x + 35 multiply to get 35 that adds up to 12 The only numbers that do both are 7 and 5. So … (x + 7)(x + 5) (x + 5)(x + 7) is also OK.
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Factor x 2 + 13x + 36
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Factor x 2 + 13x + 36 +
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Factor x 2 + 13x + 36 + There are lots of ways to get 36, like 6 6, 9 4, and 12 3.
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Factor x 2 + 13x + 36 + There are lots of ways to get 36, like 6 6, 9 4, and 12 3. Only 9 + 4 adds up to 13.
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Factor x 2 + 13x + 36 + There are lots of ways to get 36, like 6 6, 9 4, and 12 3. Only 9 + 4 adds up to 13. So the answer is (x + 9)(x + 4)
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Factor x 2 + 13x + 40 x 2 + 10x + 24 x 2 + 10x + 9
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Factor x 2 + 13x + 40 (x + 8)(x + 5) x 2 + 10x + 24 (x + 6)(x + 4) x 2 + 10x + 9 (x + 9)(x + 1)
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Factor x 2 – 16x + 48
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Factor x 2 – 16x + 48 The rule is still the same Multiply to get 48 Add to get -16
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Factor x 2 – 16x + 48 The rule is still the same -12 -4 = 48 -12 + -4 = -16
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Factor x 2 – 16x + 48 The rule is still the same -12 -4 = 48 -12 + -4 = -16 So it’s (x – 12)(x – 4).
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Factor x 2 – 5x + 6 x 2 – 16x + 55 x 2 – 18x + 32
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Factor x 2 – 5x + 6 (x – 2)(x – 3) x 2 – 16x + 55 (x – 11)(x – 5) x 2 – 18x + 32 (x – 16)(x – 2)
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Factor x 2 – x – 72
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Factor x 2 – x – 72 + This time we need both positive and negative factors because we’re multiplying to get -72. We also need to add to -1.
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Factor x 2 – x – 72 + Consider (x + 9)(x – 8) and (x – 9)(x + 8) Both multiply to -72 Only the 2 nd adds to -1 So … It’s (x – 9)(x + 8)
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Factor x 2 + 5x – 24 This time we need to multiply to -24 and add to positive 5
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Factor x 2 + 5x – 24 (x + 8)(x – 3)
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If you have one positive and one negative factor, the larger factor has the same sign as the middle term in the trinomial. x 2 + 4x – 21 = (x + 7)(x – 3) x 2 – 3x – 18 = (x – 6)(x + 3)
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Factor x 2 + 5x – 36 x 2 – 4x – 32 x 2 + 12x – 28
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Factor x 2 + 5x – 36 (x – 4)(x + 9) x 2 – 4x – 32 (x – 8)(x + 4) x 2 + 12x – 28 (x + 14)(x – 2)
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Factor x 2 + 6x + 9
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Factor x 2 + 6x + 9 +
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Factor x 2 + 6x + 9 + (x + 3)(x + 3)
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Factor x 2 + 6x + 9 + (x + 3)(x + 3) Most books would write this as (x + 3) 2
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Factor x 2 + 16x + 64 x 2 – 18x + 81 x 2 + 12x + 36
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Factor x 2 + 16x + 64 (x + 8) 2 x 2 – 18x + 81 (x – 9) 2 x 2 + 12x + 36 (x + 6) 2
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Factor x 2 – 49
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Factor x 2 – 49 We need to multiply to get -49
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0x Factor x 2 – 49 We need to multiply to get -49 We need to add to get 0
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0x Factor x 2 – 49 We need to multiply to get -49 We need to add to get 0 It’s (x + 7)(x – 7)
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Factor x 2 – 100 x 2 – 1 x 2 – 25
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Factor x 2 – 100 (x + 10)(x – 10) x 2 – 1 (x – 1)(x + 1) x 2 – 25 (x + 5)(x – 5)
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Let’s try a bit of everything.
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x 2 + 8x + 12 x 2 – x – 20 x 2 – 16x + 64 x 2 – 12x + 27
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x 2 + 8x + 12 (x + 6)(x + 2) x 2 – x – 20 (x – 5)(x + 4) x 2 – 16x + 64 (x – 8) 2 x 2 – 12x + 27 (x – 3)(x – 9)
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x 2 – 4x - 45 x 2 – 16 x 2 + 2x + 1 x 2 – 18x + 72
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x 2 – 4x - 45 (x – 9)(x + 5) x 2 – 16 (x – 4)(x + 4) x 2 + 2x + 1 (x + 1)(x + 1) x 2 – 18x + 72 (x – 12)(x – 6)
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Your book also likes problems like this. Factor a 2 + 2ab – 15b 2
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Factor a 2 + 2ab – 15b 2 The rules are still the same, but the answer will have both a and b in it.
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Factor a 2 + 2ab – 15b 2 Multiply to get -15 Add up to 2
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Factor a 2 + 2ab – 15b 2 Multiply to get -15 Add up to 2 It’s (a + 5b)(a – 3b)
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Factor x 2 + 6xy + 8y 2 n 2 – 2np – 35p 2
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Factor x 2 + 6xy + 8y 2 (x + 4y)(x + 2y) n 2 – 2np – 35p 2 (n + 5p)(n – 7p)
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Your book also likes problems like this. Factor x 10 + 16x 5 + 63
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Factor x 10 + 16x 5 + 63 What’s different this time is that the first part has x 10. This means the answer will have the form (x 5 + __)(x 5 + __)
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Factor x 10 + 16x 5 + 63 + Everything else is the same. So, the answer is … (x 5 + 9)(x 5 + 7)
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Factor x 4 – 26x 2 + 25 completely.
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Factor x 4 – 26x 2 + 25 completely. (x 2 – 25)(x 2 – 1)
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Factor x 4 – 26x 2 + 25 completely. (x 2 – 25)(x 2 – 1) … BUT, we’re not done. Both parts can be factored again.
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Factor x 4 – 26x 2 + 25 completely. (x 2 – 25)(x 2 – 1) (x + 5)(x – 5)(x + 1)(x – 1)
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There are two more things that can complicate factoring.
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First … Quadratic coefficients
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If there’s a coefficient that makes the problem look like ax 2 + bx + c The answer will usually have the form (ax + __)(x + __)
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ax 2 + bx + c You still want to find numbers that will multiply to “c”. (ax + __)(x + __)
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ax 2 + bx + c Unfortunately, they WON’T just add up to b.
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ax 2 + bx + c Remember FOIL. Outside + Inside needs to add to b
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Factor 3x 2 + 23x + 14
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Factor 3x 2 + 23x + 14 The answer will have the form (3x + __)(x + __)
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Factor 3x 2 + 23x + 14 The answer will have the form (3x + __)(x + __) Since 7 2 = 14, it might be (3x + 7)(x + 2) or (3x + 2)(x + 7)
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Factor 3x 2 + 23x + 14 Which is right? (3x + 7)(x + 2) (3x + 2)(x + 7) Check outside + inside
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Factor 3x 2 + 23x + 14 Which is right? (3x + 7)(x + 2) 6 + 7 = 13 (3x + 2)(x + 7) 21 + 2 = 23
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Factor 3x 2 + 23x + 14 The answer is (3x + 2)(x + 7)
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Factor 5x 2 + 2x – 3
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Factor 5x 2 + 2x – 3 Could be (5x + __)(x – __) or (5x – __)(x + __)
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Factor 5x 2 + 2x – 3 The numbers at the end will be 3 and 1 (one + and one –)
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Factor 5x 2 + 2x – 3 Consider (5x + 3)(x – 1) (5x + 1)(x – 3) (5x – 3)(x + 1) (5x – 1)(x + 3) Check outside + inside
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Factor 5x 2 + 2x – 3 Consider (5x + 3)(x – 1) -5+3= -2 (5x + 1)(x – 3) -15+1= -14 (5x – 3)(x + 1) 5–3 = 2 (5x – 1)(x + 3) 15–1 = 14 Check outside + inside
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Factor 5x 2 + 2x – 3 The answer is (5x – 3)(x + 1)
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Factor 2x 2 + 19x + 24 7x 2 – 37x + 10 3x 2 – x – 10
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Factor 2x 2 + 19x + 24 (2x + 3)(x + 8) 7x 2 – 37x + 10 (7x – 2)(x – 5) 3x 2 – x – 10 (3x + 5)(x – 2)
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The other possible complication is common factors.
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Common factor problems usually involve binomials, like this one: Factor 6x 7 + 15x 6
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The answer typically has the form ___( __ + __ ) The common factor goes outside the parentheses. Divide the original problem by the common factor to get what stays in the parentheses.
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Factor 6x 7 + 15x 6
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Factor 6x 7 + 15x 6 To find the common factor… Find the biggest number that goes into both 6 and 15 (the GCF) Choose the smaller exponent … Here it’s 3x 6
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Factor 6x 7 + 15x 6 So our answer has the form 3x 6 ( __ + __ )
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Factor 6x 7 + 15x 6 So our answer has the form 3x 6 ( __ + __ ) Now divide both terms by 3x 6 Divide coefficients. Subtract exponents.
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Factor 6x 7 + 15x 6 The final answer is … 3x 6 (2x + 5)
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Factor 12x 4 y 2 – 8xy 3
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Factor 12x 4 y 2 – 8xy 3 Common factor is 4xy 2 So answer is 4xy 2 (__ + __)
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Factor 12x 4 y 2 – 8xy 3 4xy 2 (3x 3 – 4y)
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Factor 7x 5 + 21x 4 18x 3 – 27x 4 30a 5 b 2 + 25a 3 b 3
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Factor 7x 5 + 21x 4 7x 4 (x + 3) 18x 3 – 27x 4 9x 3 (2 – 3x) 30a 5 b 2 + 25a 3 b 3 5a 3 b 2 (6a 2 + 5b)
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Factor 2x 4 + 16x 3 + 30x 2 completely.
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Factor 2x 4 + 16x 3 + 30x 2 completely. First take out a common factor. Here it’s 2x 2
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Factor 2x 4 + 16x 3 + 30x completely. 2x 2 (x 2 + 8x + 15) Now factor what’s inside the parentheses.
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Factor 2x 4 + 16x 3 + 30x completely. 2x 2 (x 2 + 8x + 15) = 2x 2 (x + 5)(x + 3)
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