How to Understand Factoring Polynomials
The key to factoring polynomials is an understanding of the distributive property of multiplication: a(b + c) = ab + ac. When factoring polynomials, you are trying to simplify an expression, e.g., reducing "ab + ac" to a(b + c)." This is done by observing a common element (an integer factor, or a variable that appears in every term) and factoring it out of the expression.
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Instructions
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Polynomials with Common Factors in Every Terms
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1
Check your polynomial to see if every term has a common factor. For example, consider the polynomial "6x^3 + 18x^2 -- 24x, where, for example, "6x^3" means "six multiplied by x cubed." Every term contains an "x," hence this is a common factor. Similarly, each coefficient is a multiple of 6, so this is also a common factor.
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2
Divide each term by the common factors. In our example, dividing each term by "6x" will leave us with the expression "x^2 + 3x - 4."
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3
State your answer as a product of the common factors and the expression we determined in Step 2. In our example, we would write "6x(x^2 +3x - 4)." Some factoring problems may be considered complete at this step. However, our example can be factored still further as seen in the next section.
Factoring the Product of Two Binomials
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4
Determine if your polynomial could be the product of two binomials. If your polynomial has the form "ax^2 + bx + c," where a, b and c are constants (and may be positive or negative), you may be able to factor your expression further. In our example from Section 1, the expression "x^2 + 3x - 4" has a = 1, b = 3, and c = -4.
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5
Determine the factors. A polynomial can be expressed as the product of two binomials: (dx + f)(gx + h) = dgx^2 + (dh + gf)x + hf, where d, f, g and h are all constants. You have the formulas, "dg = a," "dh + gf = b," and "fh = c" to guide your search for factors. A trial-and-error approach to the problem is usually employed. In our example, we know that d and g must both equal 1, since their product a = 1 and both are constrained to integer values. This leaves us with f and h. Since the product fh = c = -4, we know that either f or h is negative (but they cannot both be negative). We also know that f + h = 3 since b = 3. The possible values of (f, h) are (-1, 4), (-2, 2) and (1, -4). Of these three choices, only (-1, 4) consists of two numbers whose sum is 3.
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6
Write your answer as a product of factors. Our example, "x^2 +3x - 4" is reduced to "(x - 1)(x + 4)." Our original expression from Section 1, "6x^3 + 18x^2 -- 24x" would be completely factored as "6x(x - 1)(x + 4)."
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7
Check your answer. If you expand your factored expression and combine like terms, you should arrive at your original problem.
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