Efficient Techniques for Factoring Polynomials of Degree 3- A Comprehensive Guide_1

How to Factor Polynomial with Degree 3

Polynomial factorization is a fundamental skill in algebra that involves expressing a polynomial as a product of simpler polynomials. For polynomials of degree 3, factoring can be a bit more challenging than for lower-degree polynomials. However, with the right techniques and a systematic approach, factoring a polynomial with degree 3 can be achieved. In this article, we will explore various methods to factor a polynomial with degree 3, including grouping, synthetic division, and the quadratic formula.

Grouping

One of the most common methods to factor a polynomial with degree 3 is grouping. This technique involves rearranging the terms of the polynomial so that two groups of two terms can be factored separately. Here’s a step-by-step guide on how to factor a cubic polynomial using grouping:

1. Arrange the terms of the polynomial in descending order of their degrees.
2. Group the first two terms and the last two terms.
3. Factor out the greatest common factor (GCF) from each group.
4. If the GCF of the two groups is the same, factor it out as a common factor.
5. Use the quadratic formula to factor the remaining quadratic expression.

For example, consider the polynomial \(x^3 – 5x^2 + 4x – 20\). We can group the terms as follows:

\((x^3 – 5x^2) + (4x – 20)\)

Now, factor out the GCF from each group:

\(x^2(x – 5) + 4(x – 5)\)

Since the GCF of the two groups is \(x – 5\), we can factor it out:

\((x – 5)(x^2 + 4)\)

The remaining quadratic expression \(x^2 + 4\) cannot be factored further, so the final factored form of the polynomial is \((x – 5)(x^2 + 4)\).

Synthetic Division

Another method to factor a cubic polynomial is synthetic division. This technique is particularly useful when the polynomial has a linear factor. Here’s how to factor a cubic polynomial using synthetic division:

1. Arrange the polynomial in descending order of its degrees.
2. Choose a potential root for the polynomial. This can be a rational number or an irrational number.
3. Perform synthetic division using the chosen root.
4. If the remainder is zero, the chosen root is a factor of the polynomial.
5. Use the resulting quadratic expression to factor the polynomial further.

For example, consider the polynomial \(x^3 – 6x^2 + 11x – 6\). We can choose \(x = 1\) as a potential root:

\[
\begin{array}{c|ccc}
1 & 1 & -6 & 11 & -6 \\
\hline
& 1 & -5 & 6 & 0 \\
\end{array}
\]

Since the remainder is zero, \(x = 1\) is a factor of the polynomial. Now, we can factor the remaining quadratic expression \(x^2 – 5x + 6\):

\((x – 2)(x – 3)\)

Therefore, the factored form of the polynomial is \((x – 1)(x – 2)(x – 3)\).

The Quadratic Formula

The quadratic formula is a powerful tool for factoring cubic polynomials when one of the roots is a rational number. To use the quadratic formula, follow these steps:

1. Identify the quadratic expression in the factored form of the cubic polynomial.
2. Apply the quadratic formula to find the roots of the quadratic expression.
3. Use the roots to factor the cubic polynomial.

For example, consider the polynomial \(x^3 – 4x^2 + 5x – 6\). The quadratic expression in the factored form is \(x^2 – 4x + 5\). Applying the quadratic formula, we get:

\(x = \frac{-(-4) \pm \sqrt{(-4)^2 – 4(1)(5)}}{2(1)}\)

\(x = \frac{4 \pm \sqrt{16 – 20}}{2}\)

\(x = \frac{4 \pm \sqrt{-4}}{2}\)

Since the discriminant is negative, the roots are complex numbers. Therefore, the factored form of the polynomial is:

\((x – 2 + i)(x – 2 – i)(x – 3)\)

In conclusion, factoring a polynomial with degree 3 can be achieved using various methods, including grouping, synthetic division, and the quadratic formula. By understanding these techniques and applying them systematically, you can successfully factor cubic polynomials and gain a deeper understanding of algebraic expressions.