Factoring Cubic Polynomials

Before learning the process of factoring cubic polynomials, let us first recall the concept of cubic polynomials. Cubic polynomials are algebraic expressions in math with degree 3 and its standard form is ax³ + bx² + cx + d, where a, b, c, d are real numbers. Factoring cubic polynomials is a process of finding the factors of the cubic polynomials. We can find the factors of a cubic polynomial using different methods such as long division, trial and error method, etc. The factors can be linear, quadratic, or cubic (if it does not have any roots).

In this article, we will learn the process of factoring cubic polynomials using different methods and with a different number of terms. We will also solve various examples for a better understanding of the concept.

Factoring cubic polynomials is a process of expressing the cubic polynomials as a product of their factors. We can find the factors of a cubic polynomial using long division methods, algebraic identities, grouping, etc. A cubic polynomial has a standard form ax³ + bx² + cx + d, where a, b, c, d are real numbers. For factoring cubic polynomials, the prime factors of the constant term help us to find the linear factor of the polynomial. The factors of a cubic polynomial can be linear or quadratic. It is cubic if the polynomial has no roots.

The process of factoring cubic polynomials can be done using different methods. Generally, we follow the steps given below to find the factors of the cubic polynomials:

• Step 1: Find a root, say 'a', of the cubic polynomial. Then (x - a) is the factor. (This can be one of the prime factors of the constant term of the polynomial)
• Step 2: Now, divide the linear factor by the cubic polynomial to find a quadratic factor of the polynomial.
• Step 3: Factorise the quadratic polynomial obtained in step 2 using the appropriate method (grouping, splitting the middle term, algebraic identities, etc.), if possible.
• Step 4: Express the given cubic polynomial as a product of its factors.

Let us factorize a cubic polynomial using the grouping method to understand the process of factoring cubic polynomials.

Example 1: Factorize the cubic polynomial f(x) = x³ − 5x² + 4x − 20.

Solution: To factorize the polynomial f(x), we will divide it into groups.

f(x) = x³ − 5x² + 4x − 20

= (x³ − 5x²) + (4x − 20)

= x² (x - 5) + 4 (x - 5) ---- [Taking common terms out]

= (x - 5) (x² + 4)

Now, since x² + 4 does not have real roots, so we have expressed the given cubic polynomial as a product of its factors (x - 5) and (x² + 4).

The rational root theorem states that the possible roots of a cubic polynomial f(x) = ax³ + bx² + cx + d are given by ± (d/a). These roots help us to find the factors of the cubic polynomial. Let us solve an example based on the rational root theorem to understand its application.

Example: Factorize the cubic polynomial f(x) = x³ + 5x² − 2x − 24.

Solution: To find a linear factor of the polynomial, let us find the possible roots. They are ± (1, 2, 3, 4, 6, 8, 12, 24) / 1 = ± 1, 2, 3, 4, 6, 8, 12, 24. Now, let us check for each factor and find the zero of the cubic polynomial.

f(1) = 1 + 5 − 2 − 24 ≠ 0

f(-1) = −1 + 5 + 2 − 24 ≠ 0

f(2) = 8 + 20 − 4 − 24 = 0

So, x = 2 is a zero of the given polynomial which implies (x - 2) is a factor of the given cubic polynomial. Now, dividing (x - 2) by x³ + 5x² − 2x − 24, we have

(x³ + 5x² − 2x − 24) / (x - 2) = x² + 7x + 12.

Now, we will check if x² + 7x + 12 can further be factorized. We can factorize x² + 7x + 12 by splitting the term.

x² + 7x + 12 = x² + 4x + 3x + 12

= x(x + 4) + 3 (x + 4)

= (x + 3) (x + 4)

So, on factoring cubic polynomial x³ + 5x² − 2x − 24, we can express it as x³ + 5x² − 2x − 24 = (x - 2) (x + 3) (x + 4).

When a cubic polynomial has two terms, we can use an appropriate algebraic identity to factorize. If a constant term is missing in a cubic polynomial, then one of the factors is always the variable. If a constant term is there is a cubic polynomial with two terms, then we use the algebraic identities. Let us discuss the two cases:

• If constant term is missing, then a cubic polynomial with two terms can be of the form: ax³ + bx², ax³ + cx which can be factorized as ax³ + bx² = x²(ax + b) and ax³ + cx = x(ax² + c).
• If the constant term is present, then a cubic polynomial with two terms is of the form: ax³ + d. In this case, we can use any one of two algebraic identities:
  • a³ + b³ = (a + b) (a² - ab + b²)
  • a³ - b³ = (a - b) (a² + ab + b²)

Important Notes on Factoring Cubic Polynomials

• Factoring cubic polynomials is a process of expressing the cubic polynomials as a product of their factors.
• We can find the factors of a cubic polynomial using long division methods, algebraic identities, grouping, etc.

☛ Related Articles:

• Linear, Quadratic and Cubic Polynomials
• Factoring Formulas
• Factored Form

What is Factoring Cubic Polynomials?

Factoring cubic polynomials is a process of finding the factors of the cubic polynomials. In other words, we can say that factoring cubic polynomials is a process of expressing the cubic polynomials as a product of their factors.

How to Factorize Cubic Polynomials?

We can find the factors of a cubic polynomial using long division methods, algebraic identities, grouping, rational root theorem, etc.

How to Factorize Cubic Polynomials Using Factor Theorem?

We find one factor of the cubic polynomial by trial and error. Then, we can use the factor theorem to check if the root is correct or not. Then, we divide the cubic polynomial by the factor to obtain a quadratic factor. Further, we can apply the standard methods to factorize the quadratic factor into linear factors.

How to Use Rational Root Theorem for How to Factorize Cubic Polynomials?

The rational root theorem states that the possible roots of a cubic polynomial f(x) = ax³ + bx² + cx + d are given by ± (d/a). These roots help us to find the factors of the cubic polynomial.

What is How to Factorize Cubic Polynomials with Two Terms?

To factorize cubic polynomials with terms, we have two cases:

• If constant term is missing, then a cubic polynomial with two terms can be of the form: ax³ + bx², ax³ + cx which can be factorized as ax³ + bx² = x²(ax + b) and ax³ + cx = x(ax² + c).
• If the constant term is present, then a cubic polynomial with two terms is of the form: ax³ + d. In this case, we can use any one of two algebraic identities:
  • a³ + b³ = (a + b) (a² - ab + b²)
  • a³ - b³ = (a - b) (a² + ab + b²)