Factoring Polynomials | ||
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Introduction | ||
In this section, you will learn how to factor various quadratic expressions, which are 2nd degree polynomials. Here are the sections in this lesson:
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Before learning how to factor polynomials, you must first understand how to multiply polynomials. If you are not knowledgeable of polynomial multiplication, or addition and subtraction, view this lesson before proceding to the sections that follow.
esson: Operations on Polynomials
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There important reasons for learning about factors. One important reason is that large prime numbers are used for making the Internet secure. Having secure transactions over the Internet is vital for businesses, governments, and personal data. Here is Dr. Holly Krieger who explains a conjecture using a difference of two squares: Catalan's Conjecture -- Numberphile. [This is not a MATHguide source]. | ||
This video will explain how to factor polynomials which contain terms that have common factors.
ideo: Factoring: Greatest Common Factor | ||
This video will explain how to factor binomials that have the form called a difference of two squares. ideo: Factoring: Difference of Two Squares Try this interactive quizmaster to determine if you understand this lesson.
uizmaster: Factoring a Difference of Two Squares | ||
Factoring ax^{2} + bx + c when a = 1 | ||
When attempting to factor quadratics that have a leading coefficient of 1, we must focus on the values of 'b' and 'c.' The systematic approach for factoring involves factoring 'c' in as many different ways as possible into pairs. Then, find the pair that has a sum that amounts to the value of 'b.' In short, we are looking for a pair of numbers that satisfy two requirements at the same time: they must have a product equal to 'c' and a sum equal to 'b.' Use the examples below to help clarify this technique.
ideo: Factoring Trinomials | ||
Factoring ax^{2} + bx + c when a > 1 | ||
This type of trinomial is much more difficult to factor than the type 1 variety. Instead of factoring the 'c' value alone, one has to also factor the 'a' value. This compounds the number of checks necessary to verify our possible solution. As if this was not enough work, the checking method is much more involved, too. Our factors of 'a' become coefficients of our x-terms and the factors of 'c' will go right where they did for the type 1 problems, to the right of each binomial. When we place numbers to form binomials in every possible combination, we then have to check to see which factorization, if any, works. To check, multiple the first coefficient times the right-most right number to get one product and multiply the second coefficient times the left-most right number to get the second product. Add these two products and check to see if this sum is equal to 'b' in the original trinomial. This checking method is best described and understood graphically, so examine the two examples below for further clarification.
uizmaster: Factoring Complex Quadratics | ||
Factoring ax^{2} + bx + c when a < 1 | ||
It is possible to have a polynomial with a < 1, in other words with a leading coefficient less than 1. In the case that our leading coefficient is negative, simply factor out the -1 and use the techniques described above on the resulting trinomial. For example, this trinomial can be factored. |
After reading the lessons, try our videos:
ideo: Factoring: Greatest Common Factor |
After reading the lessons, try our quizmasters. MATHguide has developed numerous testing and checking programs to solidify these skills:
uizmaster: Factoring Binomials for the GCF |
MATHguide also has interactive lessons that can demonstrate the need to know how to simplify algebraic exressions:
esson: Operations on Polynomials
esson: Algebra Magic 1: Simple |