Factoring Polynomials is defined as finding factors of a polynomial into smaller non-divisible polynomials. Factorization results in the factors that when combined together, make the same polynomial. Factoring a polynomial is the opposite process of multiplying polynomials.
Any polynomial of the form F(a) can also be written as P(x) = Q(x)*D(x) + R where Dividend = Quotient * Divisor + Remainder .
If the polynomial F(x) is fully divisible by Q(x), then the remainder will be zero. Thus, F(x) = Q(x) * D(x). Thus, the polynomial F(x) is a product of two other polynomials Q(x) and D(x). There are several ways to solve the polynomial equation which depends on the terms.
Greatest Common Factor [GCF]
The first method for factoring polynomials will be factoring out the greatest common factor. When factoring in general this will also be the first thing that we should try as it will often simplify the problem. To use this method all that we do is look at all the terms and determine if there is a factor that is in common to all the terms. If there is, we will factor it out of the polynomial.
In this, we are using distributive law which is a(b+c)=ab+bc. So, finding greatest common factor we reverse the distributive law where a is common so we factor it out using the reverse distributive law as ab+ac = a(b+c).
Some of the identities used while factoring polynomials are:
- a2 − b2 = (a+b)(a−b)
- a2 + 2ab + b2 = (a+b)(a+b)
- a2 − 2ab + b2 = (a−b)(a−b)
- a3 + b3 = (a+b)(a2−ab+b2)
- a3 − b3 = (a−b)(a2+ab+b2)
- a3+3a2b+3ab2+b3 = (a+b)3
- a3−3a2b+3ab2−b3 = (a−b)3
Example: Find GCF from the following polynomial:
Solution: 1. =
Factoring Simple Polynomial
Step 1. Find common factors of GCF
Step 2. Use suitable method to solve the polynomial term.
Solution: By reversing the distributive law and findind the common factors we have, =
Differences of Two Squares
Remember the formula for solving the differences of two squares polynomial is
Solution: = (y-2) (y+2)
Solution: = (2x+9) (2x-9)
Factoring Quadratic Polynomials
It has the form such as where the coefficients a, b, and c, are real numbers. As we know that the largest exponent in a quadratic polynomial will be a 2 and by using the proper quadratic formula we can solve those equations.
Solution: This particular polynomial is factorable. First, ac=−15 and the factors of -15 are-1 are -3 and 5. So, by expanding the middle term and then group the factors.
= = x(x-3)+5(x-3) = (x-3)(x+5)
Solution: This particular polynomial is factorable. First, ac=9 and the factors of 9 are 1 are 3 and 9. So, by expanding the middle term and then grouping the factors.
= = (x+3)(x+3)
Factorise the following polynomials: