A third power polynomial, also called a cubic polynomial, includes at least one monomial or term that is cubed, or raised to the third power. An example of a third power polynomial is 4x^3 - 18x^2 - 10x. To learn how to factor these polynomials, begin by getting comfortable with three different factoring scenarios: sum of two cubes, difference of two cubes and trinomials. Then, you can move on to more complicated equations, such as polynomials with four or more terms. When you are factoring a polynomial you are essentially breaking down the equation into pieces (factors) that when multiplied will yield back the original equation.
- Skill level:
Use the standard formula a^3 + b^3 = (a+b)(a^2 - ab + b^2) when factoring an equation with one cubed term added to another cubed term, such as x^3 + 8.
Determine what represents a in the equation you are factoring. In the example x^3 + 8, x represents a, since x is the cube root of x^3.
Determine what represents b in the equation you are factoring. In the example, x^3 + 8, b^3 is represented by 8; thus, b is represented by 2, since 2 is the cube root of 8.
Factor the polynomial by filling in the values of a and b into the solution (a+b)(a^2 - ab + b^2). If a=x and b=2, then the solution is (x+2)(x^2 - 2x + 4).
Solve a more complicated equation using the same methodology. For example, solve 64y^3 + 27. Determine that 4y represents a and 3 represents b. The solution is (4y+3)(16y^2 - 12y + 9).
Use the standard formula a^3 - b^3 = (a-b)(a^2 + ab + b^2) when factoring an equation with one cubed term subtracting another cubed term, such as 125x^3 - 1.
Determine what represents a in the polynomial you are factoring. In 125x^3 - 1, 5x represents a, since 5x is the cube root of 125x^3.
Determine what represents b in the polynomial. In 125x^3 - 1, 1 is the cube root of 1, thus b = 1.
Fill in the a and b values into your factoring solution (a-b)(a^2 + ab + b^2). If a=5x and b=1, the solution is (5x-1)(25x^2 + 5x +1).
Factor a third power trinomial (a polynomial with three terms) such as x^3 + 5x^2 + 6x.
Think of a monomial that is a factor of each of the terms in your equation. In x^3 + 5x^2 + 6x, x is a common factor for each of the terms. Place the common factor outside of a pair of brackets. Divide each term of your original equation by x and place the solution inside the brackets:
x(x^2 + 5x + 6)
x^3 divided by x equals x^2, 5x^2 divided by x equals 5x and 6x divided by x equals 6.
Factor the polynomial that is inside the brackets. In the example problem, this is (x^2 + 5x + 6). Think of all the factors of 6, the last term of the polynomial. The factors of 6 are 2x3 and 1x6.
Note the centre term of the polynomial inside the brackets -- 5x in this case. Select the factors of 6 that add up to 5, the coefficient of the central term. 2 and 3 add up to 5.
Write two sets up brackets. Place x at the beginning of each bracket followed by an addition sign. Next to one addition sign write down the first selected factor (2). Next to the second addition sign write the second factor (3). It should look like this:
Remember the original common factor (x) to write your complete solution: x(x+3)(x+2)
Tips and warnings
- Check your factoring solution by multiplying the factors. If you yield the original polynomial, you factored the equation correctly.
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