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. \(\color{blue}{\:125x^3-150x^2y+60xy^2-8y^3}\)

Identity vs equation: how to tell them apart

An identity and an ordinary equation can look alike, but they behave differently.

A quick test: simplify both sides fully. If they become the same polynomial, term by term, you have an identity.

How to prove a polynomial identity

1. Pick the more complicated side.
2. Expand it using the identities you know.
3. Combine like terms.
4. Check that the result matches the other side exactly.

Example: Prove that (x + 1)³ − (x − 1)³ = 6x² + 2.

Expand each cube: (x + 1)³ = x³ + 3x² + 3x + 1 and (x − 1)³ = x³ − 3x² + 3x − 1. Subtract: (x³ + 3x² + 3x + 1) − (x³ − 3x² + 3x − 1) = 6x² + 2. The left side equals the right side, so the identity is proved.

Plugging in numbers can show that something is not an identity, but a few true checks are not a proof. You need the algebra.

Bonus, mental math: 51 × 49 = (50 + 1)(50 − 1) = 50² − 1² = 2500 − 1 = 2499.

Frequently Asked Questions

What is a polynomial identity?

A polynomial identity is an equation between two polynomial expressions that is true for every value of the variable, not just a few. For example, (x − 4)² = x² − 8x + 16 always holds. Test x = 1: the left side is (−3)² = 9 and the right side is 1 − 8 + 16 = 9.

What is a polynomial identity example?

A classic example is the difference of squares: a² − b² = (a + b)(a − b). It works for any a and b, and it even helps with mental math. To find 51 × 49, write it as (50 + 1)(50 − 1) = 50² − 1² = 2,500 − 1 = 2,499.

How do you prove a polynomial identity?

Start with the more complicated side, expand or factor it, and simplify until it matches the other side exactly. To prove (x + 1)(x² − x + 1) = x³ + 1, multiply out: x³ − x² + x + x² − x + 1. The middle terms cancel, leaving x³ + 1. Checking a few numbers is a good test but not a proof.

Which polynomial identity will prove that (3 + 6)² = 81?

Use the square of a sum: (a + b)² = a² + 2ab + b². With a = 3 and b = 6, you get 3² + 2(3)(6) + 6² = 9 + 36 + 36 = 81. That matches 9² = 81. Notice that 3² + 6² alone is only 45, which shows why the 2ab term matters.

How do you use polynomial identities to factor?

Look for an expression that matches the shape of a known identity, then write the factored side. x² − 49 fits a² − b² with b = 7, so it factors as (x − 7)(x + 7). For 8x³ − 27, use the difference of cubes with a = 2x and b = 3 to get (2x − 3)(4x² + 6x + 9).

How do you use polynomial identities to multiply?

Rewrite the numbers or expressions so they fit an identity, then use the shortcut instead of long multiplication. For 98², write (100 − 2)² = 100² − 2(100)(2) + 2² = 10,000 − 400 + 4 = 9,604. The same idea expands (2x + 5)² quickly as 4x² + 20x + 25.