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 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.
- \(\color{blue}{\left(6x\:+\:5y\right)^2\:+\:\left(6x\:-\:5y\right)^2}\)
- \(\color{blue}{\left(4x^3-3\right)^2}\)
- \(\color{blue}{\left(2x^2+y^3\right)^2\left(3x^2+y^3\right)}\)
- \(\color{blue}{\left(5x-2y\right)^3}\)
- \(\color{blue}{72x^2+50y^2}\)
- \(\color{blue}{16x^6-24x^3+9}\)
- \(\color{blue}{12x^6+16x^4y^3+4x^2y^6+3y^6x^2+y^9}\)
- \(\color{blue}{\:125x^3-150x^2y+60xy^2-8y^3}\)
Related to This Article
More math articles
- A Deep Dive into the nth Term Test for Divergence
- Top 10 Tips You MUST Know to Retake the TExES Math Test
- What Kind of Math Learner Is Your Child?
- Convert Between Fractions and Decimals
- Geometry Puzzle – Critical Thinking 19
- Graphing Histograms
- Full-Length 6th Grade SBAC Math Practice Test
- The Ultimate College Mathematics Course (+FREE Worksheets & Tests)
- How to Perform Similarity Transformations: Step-by-Step Guide
- ACT Math: Test-Taking Strategies
What people say about "Polynomial Identity - Effortless Math: We Help Students Learn to LOVE Mathematics"?
No one replied yet.