Polynomial Identity
Polynomial identity is a mathematical fact that helps us to quickly solve expressions that contain large numbers and powers. In this guide, you will learn more about polynomial identity.
![Polynomial Identity](https://www.effortlessmath.com/wp-content/uploads/2022/04/Blog-Polynomial-Identity-512x240.jpg)
Polynomial identities are equations that hold true for all possible values of the variable. When solving problems with polynomial identities, identify the pattern to see if the form is simplified or factored form, and then apply the identity and solve.
Related Topics
A step-by-step guide to polynomial identity
Polynomial identity refers to an equation that is always true regardless of the values assigned to the variables. We use polynomial identities to expand or factorize polynomials.
We must learn polynomial identities in mathematics. Four important identities of the polynomial are listed below.
- \(\color{blue}{\left(a\:+\:b\right)^2=\:a^2+\:2ab\:+\:b^2}\)
- \(\color{blue}{\left(a\:−\:b\right)^2=\:a^2−\:2ab\:+\:b^2}\)
- \(\color{blue}{\left(a\:+\:b\right)\left(a\:−\:b\right)\:=\:a^2−\:b^2}\)
- \(\color{blue}{\left(x\:+\:a\right)\left(x\:+\:b\right)\:=\:x^2+\:x\left(a\:+\:b\right)\:+\:ab}\)
Apart from the simple polynomial identities mentioned above, there are other identities of polynomials. Here are some of the most common polynomial identities used:
- \(\color{blue}{\left(a\:+\:b\:+\:c\right)^2=\:a^2+\:b^2+\:c^2+\:2ab\:+\:2bc\:+\:2ca}\)
- \(\color{blue}{\left(a\:+\:b\right)^3=\:a^3+\:3a^2b\:+\:3ab^2+\:b^3}\)
- \(\color{blue}{\left(a\:−\:b\right)^3=\:a^3−\:3a^2b+\:3ab^2−\:b^3}\)
- \(\color{blue}{\left(a\right)^3+\:\left(b\right)^3=\:\left(a\:+\:b\right)\left(a^2−\:ab\:+\:b^2\right)}\)
- \(\color{blue}{\left(a\right)^3−\:\left(b\right)^3=\:\left(a\:−\:b\right)\left(a^2+\:ab\:+\:b^2\right)}\)
- \(\color{blue}{\left(a\right)^3+\:\left(b\right)^3+\:\left(c\right)^3−\:3abc\:=\:\left(a\:+\:b\:+\:c\right)\left(a^2+\:b^2+\:c^2−\:ab\:−\:bc−ca\right)}\)
Polynomial Identity – Example 1:
Using polynomial identities, find \(\left(3x\:-2y\right)^2\).
Solution:
To solve polynomial, use this identity: \(\left(a\:−\:b\right)^2=\:a^2−\:2ab\:+\:b^2\)
Here, \(a=3x\) and \(b=2y\).
Then: \(\left(3x\:−\:2y\right)^2=\:\left(3x\right)^2−\:2\left(3x\right)\left(2y\right)+\left(2y\right)^2=\:9x^2−\:12xy\:+\:4y^2\)
Therefore, \(\left(3x\:−\:2y\right)^2=\:9x^2−\:12xy\:+\:4y^2\)
Exercises for Polynomial Identity
Simplify each expression.
- \(\color{blue}{\left(6x\:+\:5y\right)^2\:+\:\left(6x\:-\:5y\right)^2}\)
- \(\color{blue}{\left(4x^3-3\right)^2}\)
- \(\color{blue}{\left(2x^2+y^3\right)^2\left(3x^2+y^3\right)}\)
- \(\color{blue}{\left(5x-2y\right)^3}\)
![This image has an empty alt attribute; its file name is answers.png](https://www.effortlessmath.com/wp-content/uploads/2019/12/answers.png)
- \(\color{blue}{72x^2+50y^2}\)
- \(\color{blue}{16x^6-24x^3+9}\)
- \(\color{blue}{12x^6+16x^4y^3+4x^2y^6+3y^6x^2+y^9}\)
- \(\color{blue}{\:125x^3-150x^2y+60xy^2-8y^3}\)
Related to This Article
More math articles
- Exploring Geometry Fundamentals: Study of Points, Lines, and Planes
- FREE CBEST Math Practice Test
- 4th Grade Mathematics Worksheets: FREE & Printable
- Full-Length ISEE Middle Level Math Practice Test
- FREE 6th Grade Common Core Math Practice Test
- 10 Most Common SSAT Middle Level Math Questions
- Top 10 Tips to Overcome ACT Math Anxiety
- 5th Grade MEA Math Worksheets: FREE & Printable
- How is the FTCE General Knowledge Test Scored?
- How to Master Probability Density Functions
What people say about "Polynomial Identity - Effortless Math: We Help Students Learn to LOVE Mathematics"?
No one replied yet.