How to Factor Third Power Polynomials

How to Factor Third Power Polynomials
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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 move on to more complicated equations, such as polynomials with four or more terms. Factoring a polynomial requires breaking down the equation into pieces (factors) that when multiplied will yield back the original equation.

Factor Sum of Two Cubes

    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 ​x3+ 8.

    Determine what represents ​a​ in the equation. In the example ​x3+ 8, ​x​ represents ​a​, since ​x​ is the cube root of ​x3.

    Determine what represents ​b​ in the equation. In the example, ​x3+8, ​b3 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 4​y​ represents ​a​ and 3 represents ​b​. The solution is

    (4y+3)(16y^2-12y+9)

Factor Difference of Two Cubes

    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. In 125​x3 − 1, 5​x​ represents ​a​, since 5​x​ is the cube root of 125​x3.

    Determine what represents ​b​ in the polynomial. In 125​x3 − 1, 1 is the cube root of 1, thus ​b​ = 1.

    Fill in the ​a​ and ​b​ values into the factoring solution

    (a-b)(a^2+ab+b^2)

    If ​a​ = 5​x​ and ​b​ = 1, the solution becomes

    (5x-1)(25x^2+5x+1)

Factor a Trinomial

    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 the 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 the original equation by ​x​ and place the solution inside the brackets:

    x(x^2+5x+6)

    Mathematically, ​x3 divided by ​x​ equals ​x2, 5​x2 divided by ​x​ equals 5​x​ and 6​x​ divided by ​x​ equals 6.

    Factor the polynomial inside the brackets. In the example problem, the polynomial is

    (x^2+5x+6)

    Think of all the factors of 6, the last term of the polynomial. The factors of 6 equal 2 × 3 and 1 × 6.

    Note the center term of the polynomial inside the brackets – 5​x​ 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 of 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:

    (x+3)(x+2)

    Remember the original common factor (​x​) to write the complete solution:

    x(x+3)(x+2)

    Tips

    • Check the factoring solution by multiplying the factors. If the multiplication yields the original polynomial, the equation was factored correctly.

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