How to Factor Polynomials With Fractional Coefficients

Solving polynomials can get messy if you don't eliminate fractional coefficients
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Factoring polynomials with fractional coefficients is more complicated than factoring with whole number coefficients, but you can easily turn every fractional coefficient in your polynomial into a whole number coefficient without changing the overall polynomial. Simply find a common denominator for all the fractions, and then multiply the entire polynomial by that number. This will allow you to cancel out the denominator in each fraction, leaving only whole number coefficients. You can then factor it using normal procedures for factoring.

    Find the prime factorization of the denominator of each of your fractional coefficients. The prime factorization of a number is the unique set of prime numbers that, when multiplied together, equal the number. For example, the prime factorization of 24 is 2_2_2_3 (not 2_3_4 or 8_3 because 4 and 8 aren't prime). An easy way to find the prime factorization is to repeatedly divide the number into factors until you are left with only primes: 24 = 4_6 = (2_2) * (2_3) = 2_2_2_3.

    Draw a Venn Diagram representing each of your denominators. For example, if you had three denominators, you would draw three circles, each circle slightly overlapping the other and all three overlapping in the center (see Resources: Venn Diagram for a picture). Label the circles "1," "2," etc. based on the order of the fractions in the polynomial.

    Place the prime factors in the Venn Diagram according to which denominators have them. For example, if your three denominators are 8, 30 and 10, the first has a prime factorization of (2_2_2), the second has (2_3_5), and the third has (2*5). You would put "2" in the center, because all three denominators share the factor of 2. You would put one "5" in the overlap between circle 2 and circle 3 because the second and third denominators share this factor. Finally, you would put "2" twice in the area of circle 1 with no overlap and a "3" in the area of circle 2 with no overlap, because these factors are not shared by any other denominator.

    Multiply all the numbers in your Venn Diagram to find the lowest common denominator of your fractional coefficients. In the above example, you would multiply 2 times 5 times 2 times 2 times 3 to get 120, which is the lowest common denominator of 8, 30 and 10.

    Multiply the entire polynomial by the common denominator, distributing it to each fractional coefficient. You will be able to cancel out the denominator in each coefficient, leaving only whole numbers. For example: 120(1/8_x^2 + 7/30_x + 3/10) = 15x^2 + 28x + 36.

    Write two sets of parentheses, with the first term of both sets a factor of the leading coefficient. For example, 15x^2 factors to 3x and 5x: (3x....)(5x....).

    Find two numbers that multiply together to equal your constant from the polynomial. For example, 6 times 6 or 9 times 4 equals 36. Plug them into your parentheses and see if they work: (3x + 6)(5x +6); (3x + 9) (5x + 4); (3x + 4)(5x + 9). Check your result by using FOIL to re-expand your polynomial: (3x + 4)(5x + 9) = 15x^2 + 27x + 20x +36 = 15x^2+ 47x + 36, which is not the same as our original polynomial.

    Continue plugging in different numbers until the result matches the original polynomial when re-expanded. You may need to change the first terms to different factors of the leading coefficient.

    Divide your factored polynomial by the common denominator from Step 4 to cancel out the change you made by multiplying in Step 5.

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