Polynomials are fundamental mathematical expressions that are an integral part of algebra. They are versatile, finding applications in various areas of science, engineering, and everyday problem-solving. Understanding polynomials and mastering the art of factoring them is a valuable skill that can simplify complex equations and help solve real-world problems. In this tutorial, we will cover step-by-step guide on how to factor Polynomials and these points:
- What is a polynomial? What are terms?
- How to factor polynomials with 2 terms (binomial)?
- How to factor polynomials with 3 terms (trinomials) when a=1?
- How to factor polynomials with 3 terms (trinomials) when a≠1?
- How to factor polynomials with 4 terms?
- How to factor Cubic Polynomials by grouping?
What is a polynomial?
A polynomial is a mathematical expression consisting of variables, coefficients, and exponents, combined using addition, subtraction, and multiplication. It typically takes the
integer denoting the degree of the polynomial.
For example,
In this polynomial:
- P(x) is the expression itself.
- 3,−2,5, and −7 are coefficients.
- x is the variable.
- The highest power of xx in this polynomial is 3, so it’s a third-degree polynomial.
Polynomials can come in various forms and degrees, but they all follow a similar structure of combining coefficients and powers of a variable.
When it comes to factoring polynomials, you will most commonly be dealing with polynomials that have 2 terms, 3 terms, or 4 terms:
How to a Factor Polynomials with 2 Terms
Factoring polynomials with 2 terms, which are also known as binomials, is a fundamental algebraic skill. The goal is to express the binomial as a product of two simpler expressions. Here’s a step-by-step guide on how to factor binomials:
Step 1: Identify the Greatest Common Factor (GCF)
The first step in factoring a binomial is to look for the greatest common factor (GCF) of the two terms. The GCF is the largest expression that divides both terms evenly. For example, if you have the binomial (6x + 9), the GCF is 3, which can be factored out from both terms.
Step 2: Write the GCF Outside the Parentheses
After identifying the GCF, write it outside the parentheses. Using the example from step 1, you would have 3(x + 3).
Step 3: Divide the Binomial by the GCF
To find what goes inside the parentheses, divide each term of the original binomial by the GCF. In our example, you divide both 6x and 9 by 3 to get and 3.
Step 4: Simplify the Expression Inside the Parentheses
Simplify the expression inside the parentheses, if possible. In this case, 2x+3 cannot be simplified further.
Step 5: Write the Factored Form
The factored form of the binomial is now . This is the fully factored expression.
So, the factors of 6x + 9 are 3 and (2x+3).
That’s how you factor a binomial. It’s a straightforward process of identifying the GCF, dividing both terms by it, and writing the GCF outside the parentheses with the simplified expression inside. This technique is essential for simplifying expressions and solving equations in algebra.