How to Factor Higher Degree Polynomials
Polynomial factorization is a fundamental concept in algebra that plays a crucial role in various mathematical applications. While factoring quadratic polynomials is relatively straightforward, the process becomes more complex as the degree of the polynomial increases. In this article, we will explore different methods to factor higher degree polynomials, including synthetic division, grouping, and the rational root theorem.
1. Synthetic Division
Synthetic division is a technique used to divide a polynomial by a linear factor. To factor a higher degree polynomial using synthetic division, follow these steps:
1. Identify the linear factor you want to divide by, such as (x – a).
2. Write the coefficients of the polynomial in descending order, with the leading coefficient on the left.
3. Write the divisor (a) to the left of the coefficients.
4. Bring down the first coefficient and multiply it by the divisor.
5. Add the product to the next coefficient and write the result below.
6. Repeat steps 4 and 5 until you reach the last coefficient.
7. The last number in the bottom row is the remainder, and the coefficients in the top row represent the quotient.
For example, to factor the polynomial x^3 – 3x^2 + 2x – 6 by (x – 2) using synthetic division:
“`
2 | 1 -3 2 -6
| 2 2 2
———————
| 1 -1 4 0
“`
The quotient is x^2 – x + 4, and the remainder is 0. Therefore, the factorization is (x – 2)(x^2 – x + 4).
2. Grouping
Grouping is another method used to factor higher degree polynomials. This technique involves grouping the polynomial into two or more pairs of terms and factoring out the greatest common factor (GCF) from each pair. Then, factor out the GCF of the entire polynomial.
For example, to factor the polynomial x^3 – 5x^2 + 6x – 10:
1. Group the polynomial into two pairs: (x^3 – 5x^2) + (6x – 10).
2. Factor out the GCF from each pair: x^2(x – 5) + 2(x – 5).
3. Factor out the GCF of the entire polynomial: (x – 5)(x^2 + 2).
3. The Rational Root Theorem
The rational root theorem is a useful tool for finding rational roots of a polynomial. It states that if a polynomial has a rational root p/q (where p and q are integers with no common factors), then p must be a factor of the constant term, and q must be a factor of the leading coefficient.
To factor a polynomial using the rational root theorem, follow these steps:
1. List all possible rational roots using the rational root theorem.
2. Test each possible root by synthetic division or substitution.
3. Once you find a root, factor out the corresponding linear factor.
4. Use polynomial long division or synthetic division to divide the polynomial by the linear factor.
5. Repeat steps 2-4 until the polynomial is completely factored.
By applying these methods, you can factor higher degree polynomials with confidence. Remember to practice and become familiar with each technique, as they can be applied to various polynomial problems.
