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Polynomials are mathematical expressions that consist of constants, variables, and exponentials. There are various methods to decompose or simplify these polynomials, with one of the most common techniques being factorization. Factoring allows us to break down an expression into its constituent parts, making it easier to understand and manipulate. In this guide, we will explore the process of factorizing third degree polynomials, also known as cubic polynomials. By following these step-by-step instructions, we will learn how to factorize these complex expressions, providing a deeper understanding of their structure and helping us to solve equations more efficiently.
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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
![Image titled Factor a Cubic Ppynomial Step 1](https://www.wikihow.com/images_en/thumb/1/16/Factor-a-Cubic-Ppynomial-Step-1-Version-5.jpg/v4-728px-Factor-a-Cubic-Ppynomial-Step-1-Version-5.jpg)
- Assume “consider polynomials.” x 3 + 3x 2 – 6x – 18 = 0. We group the polynomial into two parts (x 3 + 3x 2 ) and (- 6x – 18).
![Image titled Factor a Cubic Ppynomial Step 2](https://www.wikihow.com/images_en/thumb/c/cf/Factor-a-Cubic-Ppynomial-Step-2-Version-5.jpg/v4-728px-Factor-a-Cubic-Ppynomial-Step-2-Version-5.jpg)
- 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.
![Image titled Factor a Cubic Ppynomial Step 3](https://www.wikihow.com/images_en/thumb/5/5b/Factor-a-Cubic-Ppynomial-Step-3-Version-5.jpg/v4-728px-Factor-a-Cubic-Ppynomial-Step-3-Version-5.jpg)
- 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).
![Image titled Factor a Cubic Ppynomial Step 4](https://www.wikihow.com/images_en/thumb/d/db/Factor-a-Cubic-Ppynomial-Step-4-Version-5.jpg/v4-728px-Factor-a-Cubic-Ppynomial-Step-4-Version-5.jpg)
- We have (x + 3)(x 2 – 6).
![Image titled Factor a Cubic Ppynomial Step 5](https://www.wikihow.com/images_en/thumb/2/2c/Factor-a-Cubic-Ppynomial-Step-5-Version-5.jpg/v4-728px-Factor-a-Cubic-Ppynomial-Step-5-Version-5.jpg)
- The roots of the polynomial under consideration are -3, √6 and -√6.
Factoring using the free term
![Image titled Factor a Cubic Ppynomial Step 6](https://www.wikihow.com/images_en/thumb/2/2b/Factor-a-Cubic-Ppynomial-Step-6-Version-5.jpg/v4-728px-Factor-a-Cubic-Ppynomial-Step-6-Version-5.jpg)
- For example, consider the formula x 3 – 4x 2 – 7x + 10 = 0.
![Image titled Factor a Cubic Ppynomial Step 7](https://www.wikihow.com/images_en/thumb/3/37/Factor-a-Cubic-Ppynomial-Step-7-Version-5.jpg/v4-728px-Factor-a-Cubic-Ppynomial-Step-7-Version-5.jpg)
- 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.
![Image titled Factor a Cubic Ppynomial Step 8](https://www.wikihow.com/images_en/thumb/e/ed/Factor-a-Cubic-Ppynomial-Step-8-Version-5.jpg/v4-728px-Factor-a-Cubic-Ppynomial-Step-8-Version-5.jpg)
- 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.
![Image titled Factor a Cubic Ppynomial Step 9](https://www.wikihow.com/images_en/thumb/5/59/Factor-a-Cubic-Ppynomial-Step-9-Version-5.jpg/v4-728px-Factor-a-Cubic-Ppynomial-Step-9-Version-5.jpg)
- “x = 1” is equivalent to “x – 1 = 0” or “(x – 1)”. That is, we have subtracted 1 on both sides of the equation.
![Image titled Factor a Cubic Ppynomial Step 10](https://www.wikihow.com/images_en/thumb/f/f7/Factor-a-Cubic-Ppynomial-Step-10-Version-5.jpg/v4-728px-Factor-a-Cubic-Ppynomial-Step-10-Version-5.jpg)
- 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.
![Image titled Factor a Cubic Ppynomial Step 11](https://www.wikihow.com/images_en/thumb/8/80/Factor-a-Cubic-Ppynomial-Step-11-Version-5.jpg/v4-728px-Factor-a-Cubic-Ppynomial-Step-11-Version-5.jpg)
- 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).
![Image titled Factor a Cubic Ppynomial Step 12](https://www.wikihow.com/images_en/thumb/8/8b/Factor-a-Cubic-Ppynomial-Step-12-Version-5.jpg/v4-728px-Factor-a-Cubic-Ppynomial-Step-12-Version-5.jpg)
- (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 is an essential skill in algebraic problem-solving. Although it may seem complex at first, the process becomes more manageable with practice and a systematic approach. By using techniques such as factoring by grouping, applying the sum and product rules, or utilizing special cases like the difference of cubes or perfect square trinomials, one can successfully factorize third-degree polynomials. This ability is crucial in solving equations, graphing functions, and understanding the behavior of polynomials. Moreover, factoring third-degree polynomials opens doors to further mathematical concepts, such as finding roots, determining critical points, and analyzing the behavior of functions. With the right strategies, perseverance, and mathematical understanding, anyone can become proficient in factorizing third-degree polynomials.
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