Polynomial Identity

A polynomial identity is an equation between polynomials that holds for every value of the variable, such as (a + b)² = a² + 2ab + b² or the difference of squares. Below: how to spot the structure before expanding, prove an identity by matching coefficients, and the common expansion errors.

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

• How to Add and Subtract Polynomials
• How to Simplify Polynomials

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:

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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.

1. \(\color{blue}{\left(6x\:+\:5y\right)^2\:+\:\left(6x\:-\:5y\right)^2}\)
2. \(\color{blue}{\left(4x^3-3\right)^2}\)
3. \(\color{blue}{\left(2x^2+y^3\right)^2\left(3x^2+y^3\right)}\)
4. \(\color{blue}{\left(5x-2y\right)^3}\)

1. \(\color{blue}{72x^2+50y^2}\)
2. \(\color{blue}{16x^6-24x^3+9}\)
3. \(\color{blue}{12x^6+16x^4y^3+4x^2y^6+3y^6x^2+y^9}\)
4. \(\color{blue}{\:125x^3-150x^2y+60xy^2-8y^3}\)