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Terms with Common Factors |
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Definition
When factoring, we always try to factor out the largest factor common to all term in the polynomial, using the distributive property. e.g., 15 + 10x - 5x2 = 5(3 + 2x - x2) |
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Definition
In some polynomials, pairs of terms have a common binomial factor that can be removed in a process called factoring by grouping. x3 + 3x2 - 5x - 15 = (x3 + 3x2) + (-5x - 15) = x2(x + 3) - 5(x + 3) = (x + 3) (x2 - 5) |
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Trinomials of the Type x2 + bx + c |
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Definition
To factor a trinomial of the type x2 + bx + c, we look for binomial factors of the type (x + p) (x + q), where p · q = c and p + q = b. |
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Trinomials of the Type ax2 + bx + c, a ≠ 1 |
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Definition
If poss., factor out the largest common factor. Multiply the leading coefficient and the constant. Try to factor so that the sum of the factors is the coefficent of the middle term. Split the middle term using the numbers found in step 3. Factor by grouping.
12x3 + 10x2 - 8x
1. 2x(6x2 +5x -4)
2. 6(-4) = -24
3. -3 · 8 = -24; -3 + 8 = 5
4. 5x = -3x + 8x
5. 6x2 - 3x + 8x - 4
3x(2x -1) + 4(2x -1) (2x -1) (3x + 4)
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Special Factorizations: Trinomial Squares |
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Definition
A2 + 2AB + B2 = (A + B)2 A2 - 2AB + B2 = (A - B)2 |
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Special Factorizations: Sum of Cubes |
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Definition
A3 + B3 = (A + B) (A2 - AB + B2) |
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Special Factorizations: Difference of Cubes
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Definition
A3 - B3 = (A - B) (A2 + AB + B2) |
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Special Factorizations: Difference of Squares |
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Definition
A2 - B2 = (A + B) (A - B) |
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Special Factorizations: Sum of Squares |
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Definition
A2 + B2 cannot be factored using real-number coefficients. |
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Definition
- Always factor out the largest common factor first.
- Look at the number of terms
- Two terms: Try factoring as a difference of squares first. Next try factoring as a sum or a difference of cubes. Do not try to factor as a sum of squares.
- Three Terms: Try factoring as the square of a binomial. Next, try using the grouping method for factoring a trinomial.
- Four or more terms: Try factoring by grouping and factoring out a common binomial factor.
- Always factor completely. If a factor with more than one term can itself be factored further, do so.
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