How to Factor Messier Polynomials

There are many techniques for factoring more complicated polynomials.

Factoring polynomials can be a laborious process, especially if the degree (highest exponent in the polynomial) is greater than two. There are tricks to factoring polynomials that don't involve the use of a sophisticated calculator or computer, however. Any quadratic equation (polynomial of degree two) can be factored using the quadratic formula. Sums and differences of cubes can always be factored by hand. Finally, polynomials with four terms can occasionally be factored by grouping.

Instructions

Factoring Quadratic Equations

1. Convert the polynomial into the Standard Form for quadratic equations by ordering the terms by decreasing exponent from left to right: "Ax^2 + Bx + C."

2. Plug "A," "B" and "C" into the equation "B^2 - (4*A*C)" to find the discriminant. For the equation 2x^2 + 6x - 7, "A" = 2, "B" = 15, and "C" = -7 (negative because you are subtracting 7). The discriminant is then equal to 6^2 - 4*2*(-7), or 36 + 56, or 92.

3. Take the square root of the discriminant. If it's a positive square, you will be left with a positive value. If it's a positive non-square, you will be left with a radical. If it's a negative number, you will be left with an imaginary number. Simplify the radical or imaginary number by factoring out any squares. For example: SqRt(92) = SqRt(4) * SqRt (23) = 2 * SqRt(23). For negatives, SqRt(-72) = SqRt(36) * SqRt(2) * SqRt(-1) = 3*SqRt(2)*i (the square root of negative one is written as i).

4. Add (-B) to the discriminant, then divide by (2*A). This is one of your two factors.

5. Subtract the discriminant from (-B), then divide by (2*A). This is your other factor.

6. Rewrite the polynomial as (x - "First Factor")(x - "Second Factor"). For example, if your factors were 14 and -5, you would rewrite the polynomial as (x - 14)(x + 5).

Sum and Difference of Cubes

1. Write two sets of parentheses, with room for two terms in the first set and three terms in the second. In the first set, put the cube root of the first term in the polynomial, followed by a space, followed by the cube root of the second term. For example, for x^3 - 27, you would write (x 3)( ).

2. Square the first and second terms in the first set of parentheses. Write these values as the first and last terms in the second set of parentheses: (x 3)(x^2 9).

3. Multiply the first and second terms in the first set of parentheses. Write this value as the middle term in the second set of parentheses: (x 3)(x^2 3x 9)

4. Use the mnemonic "SOAP" to determine the signs in between the terms. The first sign is "Same," or the same as the original polynomial. The second sign is "Opposite," or the opposite sign as the original polynomial. The third sign is "Always Positive." For "x^3 - 27," the order of the signs would be negative, positive, positive, or (x - 3)(x^2 + 3x + 9).

Factoring by Grouping

1. Divide the polynomial into two groups, A and B, each with the same number of terms. A polynomial with four terms should be grouped into two groups of two terms, etc. Make sure the difference between the degree of the first and second terms is the same for both groups. For example: 2x^3 +3x^2 - 6x - 9 should be divided into group A = 2x^3 - 6x (difference between degrees is two) and group B: 3x^2 - 9 (difference between degrees is also two).

2. Factor both Groups by pulling out a common factor between the terms. For example:

   2x^3 - 6x = 2x(x^2 - 3)

   3x^2 - 9 = 3(x^2 - 3)

3. Recombine the two groups. The first parentheses should contain the factor that group A and B share ("x^2 - 3" in the above example). The second parentheses should contain the common factor in group A plus the common factor in group B ("2x + 3" in the above example). 2x^3 + 3x^2 - 6x - 9 = (x^2 - 3)(2x - 3).

Tips & Warnings

• Factoring by sum or difference of cubes only works if both terms are perfect cubes.

• Factoring by grouping only works if you can find a common factor between group A and B.

References

• Algebra Lab; Factoring Polynomials
• Purple Math: The Quadratic Formula Explained