Factoring PolynomialsFactoring polynomials is the method of simplifying a polynomial by converting it into the product of two or more factors. This method is helpful in solving complex problems in the later grades. Here you will learn about the generic approach to factorize a polynomial. As you must be knowing by now that a polynomial can take up various forms. Therefore in this article we will take up some of the most common types of polynomials that you might face in your curriculum and sort out a process for factorizing them. Remember, the process of factoring polynomials is about finding simpler polynomials that can be multiplied to give the polynomial we have. So let’s have a look at different ways of factoring polynomials Case 1 Common Factor In this type of polynomials you will find a common factor easily in the terms of the polynomial. The form it takes is as follows ab + ac As can be seen the term “a” is common in both the terms. The polynomial can be factorized by taking “a” as a factor. Thus the polynomial can be expressed as a(b + c) Example 3x^2 + 6x In this example you can easily see that the common expression in the terms is 3x. Thus the polynomial can be expressed as 3x (x + 2) Thus the polynomial has been factorized following the common factor method. Case 2 Difference of Squares This type of polynomials takes up the following form a^2 – b^2 This form of polynomials can be factorized in accordance with the following rule a^2 – b^2 = (a + b)(a – b) Example x^2 – 16 Now the polynomial above is in a form that can be expressed as difference of two squares. 16 can be written as square of 4. Thus the polynomial becomes x^2 – 4^2 If you compare this expression with the one given above then x = a and 4 = b. Thus x^2 – 4^2 = (x + 4)(x – 4) Therefore the two factors of the polynomial are (x + 4) and (x – 4). Case 3 Quadratic Trinomial The form of polynomial in this case is ax^2 + bx + c Before getting onto factoring polynomials I wish to discuss some of the points that you must understand regarding these polynomials. Here they are
Let’s start factoring polynomials The following rules will help you breeze through the factorization process for ax^2 + bx + c
Example x^2 + 9x + 20 In the above polynomial the value of a is 1. Thus we need to figure out two number which when multiplied gives 20 and when added to each other yield 9. Let’s start by looking for two numbers that yield 20. The possible combinations are as below
Now let us check these options for the sum of 9. Of the three combinations we have above (4, 5) is the one in which the two number when added yield 9. Thus the two numbers are 4 and 5. Next step is to arrange the polynomial such that these two numbers are accommodated into it. Follow the steps below x^2 + 9x + 20 = x^2 + (4 + 5) x + 20 = x^2 + 4x + 5x + 20 =( x^2 + 4x) + (5x + 20) Now we see that the polynomial is separated into two different expressions that are added to each other. In the first expression there are two terms with x as the common factor. Whereas in the second expression there are two terms with 5 as the common factor. Thus the polynomial can be written as =( x^2 + 4x) + (5x + 20) = x (x + 4) + 5 (x + 4) = (x + 4)*(x + 5) That’s it. You have got the factors for the polynomial x^2 + 9x + 20 There are quite a number of variations of this method. We will look into it in detail in other lessons.
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