Mastering Polynomial Factorization- A Comprehensive Guide to Factoring Quadratic Polynomials of Degree 4
How to Factorise a Polynomial of Degree 4
Polynomial factorisation is a fundamental skill in algebra, especially when dealing with polynomials of higher degrees. One common scenario is factorising a polynomial of degree 4, which can be quite challenging for many students. In this article, we will discuss various methods and techniques to factorise a polynomial of degree 4, including grouping, synthetic division, and the quadratic formula.
Grouping
Grouping is a simple and straightforward method for factorising a polynomial of degree 4. The idea is to group the terms of the polynomial into two pairs and factorise each pair separately. Then, we can combine the factors to obtain the final factorisation.
For example, consider the polynomial \(x^4 – 5x^2 + 6\). We can group the terms as follows:
\[
(x^4 – 2x^2) – (3x^2 – 6)
\]
Now, factorise each pair:
\[
x^2(x^2 – 2) – 3(x^2 – 2)
\]
Observe that both pairs have a common factor of \(x^2 – 2\). We can factor this out to get:
\[
(x^2 – 2)(x^2 – 3)
\]
Finally, we can factorise the quadratic expressions \(x^2 – 2\) and \(x^2 – 3\) further if needed.
Synthetic Division
Synthetic division is another method for factorising a polynomial of degree 4. This technique is particularly useful when we know one of the roots of the polynomial. By using synthetic division, we can quickly find the other roots and factorise the polynomial accordingly.
Let’s consider the polynomial \(x^4 – 6x^3 + 11x^2 – 6x + 6\). Suppose we know that \(x = 1\) is a root of the polynomial. We can use synthetic division to find the other roots as follows:
\[
\begin{array}{c|ccccc}
1 & 1 & -6 & 11 & -6 & 6 \\
\hline
& 1 & -5 & 6 & 0 & 6 \\
\end{array}
\]
The synthetic division table shows that \(x = 1\) is a root, and the quotient is \(x^3 – 5x^2 + 6\). We can now factorise the quotient further:
\[
x^3 – 5x^2 + 6 = (x – 2)(x^2 – 3x – 3)
\]
Therefore, the factorisation of the original polynomial is:
\[
(x – 1)(x – 2)(x^2 – 3x – 3)
\]
The Quadratic Formula
The quadratic formula is a powerful tool for factorising polynomials of degree 4. If we can express the polynomial as a product of two quadratic expressions, we can use the quadratic formula to find the roots of each quadratic and, consequently, factorise the polynomial.
For example, consider the polynomial \(x^4 – 8x^3 + 17x^2 – 12x + 4\). We can express it as the product of two quadratic expressions:
\[
(x^2 – 4x + 1)(x^2 – 4x + 4)
\]
Now, we can use the quadratic formula to find the roots of each quadratic expression:
\[
x^2 – 4x + 1 = 0 \quad \Rightarrow \quad x = \frac{4 \pm \sqrt{12}}{2} = 2 \pm \sqrt{3}
\]
\[
x^2 – 4x + 4 = 0 \quad \Rightarrow \quad x = \frac{4 \pm \sqrt{16}}{2} = 2
\]
Hence, the factorisation of the original polynomial is:
\[
(x – (2 – \sqrt{3}))(x – (2 + \sqrt{3}))(x – 2)^2
\]
In conclusion, factorising a polynomial of degree 4 can be achieved using various methods, including grouping, synthetic division, and the quadratic formula. Each method has its own advantages and can be applied depending on the specific polynomial and the available information. By mastering these techniques, students can gain a deeper understanding of polynomial factorisation and its applications in algebra and beyond.
