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Factoring third degree polynomials is an essential skill in algebraic mathematics. It involves breaking down a polynomial equation of degree three into its factors, which are then used to solve equations or simplify complex expressions. Understanding the process of factorizing third degree polynomials allows us to study the roots of the equation and analyze its behavior. In this guide, we will explore the various techniques and strategies involved in factorizing third degree polynomials, providing step-by-step instructions and examples to enhance our understanding. Whether we are students looking to master this fundamental concept or individuals seeking to refresh our knowledge, this guide will serve as a comprehensive resource to demystify the process of factorizing third degree polynomials.
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This article will show you how to factor a 3rd degree polynomial. We will learn how to factorize using the common factorization method and the method of using the free terms.
Steps
Factoring using the group method
- Assume “consider polynomials.” x 3 + 3x 2 – 6x – 18 = 0. We group the polynomial into two parts (x 3 + 3x 2 ) and (- 6x – 18).
- In the group (x 3 + 3x 2 ), we can easily see that x 2 is a common factor.
- In the group (- 6x – 18), -6 is the common factor.
- Draw x 2 as the common factor of the first group, we get x 2 (x + 3).
- Draw -6 out of the second group, we get -6(x+3).
- We have (x + 3)(x 2 – 6).
- The roots of the polynomial under consideration are -3, √6 and -√6.
Factoring using the free term
- For example, consider the formula x 3 – 4x 2 – 7x + 10 = 0.
- Multipliers of a number are numbers that we can multiply by another number to get another number. In this case, the factors of 10, or “d,” are: 1, 2, 5, and 10.
- Try with the first factor, 1. Substitute “1” into all the “x” variables in the polynomial:
(1) 3 – 4(1) 2 – 7(1) + 10 = 0 - We get: 1 – 4 – 7 + 10 = 0.
- Since 0 = 0, we have x = 1 as a solution of the equality.
- “x = 1” is equivalent to “x – 1 = 0” or “(x – 1)”. That is, we have subtracted 1 on both sides of the equation.
- Can we split (x – 1) from x 3 ? The answer is no. However, we can borrow -x 2 from the second variable and do the factorization as follows: x 2 (x – 1) = x 3 – x 2 .
- Can we separate (x – 1) from the rest of the second variable? Again the answer is no. We need to borrow part of the third variable. Taking 3x from -7x and grouping the common factor with the rest of the second variable, we get -3x(x – 1) = -3x 2 + 3x.
- Since we borrowed 3x from -7x, so the second variable will become -10x, notice the free term is 10. Can we factor it? The answer is yes: -10(x – 1) = -10x + 10.
- We have separated and rearranged the variables so that we can group (x – 1) as a common term for the whole expression. In general, we have the expression after splitting x 3 – x 2 – 3x 2 + 3x – 10x + 10 = 0, this expression is also equivalent to x 3 – 4x 2 – 7x + 10 = 0.
- x 2 (x – 1) – 3x(x – 1) – 10(x – 1) = 0. We can rearrange this equality to make it easier to factorize: (x – 1)(x 2 – 3x – 10) = 0.
- At this point, we need to factorize for (x 2 – 3x – 10). This expression can be decomposed into (x + 2)(x – 5).
- (x – 1)(x + 2)(x – 5) = 0, that is, 1, -2 and 5 are the roots of the polynomial.
- Substituting -2 into the original equation we get: (-2) 3 – 4(-2) 2 – 7(-2) + 10 = -8 – 16 + 14 + 10 = 0.
- Substituting 5 into the original equation, we get (5) 3 – 4(5) 2 – 7(5) + 10 = 125 – 100 – 35 + 10 = 0.
Advice
- For real numbers, there is no third degree polynomial that cannot be factored because all cubics have at least one real root. For polynomials that do not have a suitable real root, for example x^3 + x + 1, we cannot decompose into polynomials if we use real numbers with rational coefficients. Although it is possible to calculate the solution of this polynomial according to the cubic equation solution formula, it cannot itself be decomposed into integer polynomials.
- A third degree polynomial is the product of three first degree polynomials or the product of a first degree polynomial and a second degree polynomial that cannot be factored. In this case, after we have found the first common factor, we can do polynomial division by the polynomial to find the quadratic polynomial.
wikiHow is a “wiki” site, which means that many of the articles here are written by multiple authors. To create this article, 24 people, some of whom are anonymous, have edited and improved the article over time.
This article has been viewed 130,925 times.
This article will show you how to factor a 3rd degree polynomial. We will learn how to factorize using the common factorization method and the method of using the free terms.
In conclusion, factorizing third degree polynomials can be a complex and sometimes time-consuming process. However, by understanding the various techniques and strategies available, such as the Rational Root Theorem, synthetic division, and factoring by grouping, it becomes possible to factorize these polynomials effectively. It is important to stay focused and patient while attempting to factorize third degree polynomials as it requires careful analysis and manipulation of terms. Furthermore, practice and familiarity with the concept is crucial in order to efficiently identify patterns and successfully factorize the polynomial. Overall, factorizing third degree polynomials is a valuable skill to have in mathematics as it aids in simplifying equations and solving equations to determine the roots of the polynomial.
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