Proving Polynomial Identities

This segment will provide a proof of the most commonly used four polynomial identities which are:

• (a+b)² = a²+b²+2ab
• (a-b)² = a²+b²-2ab
• (a+b)(a-b)² = a²-b²
• (x + a)(x + b) = x² + x(a + b) + ab

Identity 1: (a+b)² = a²+b²+2ab

The expression (a+b)² can be expanded using the distributive property:

(a+b)² = (a+b) · (a+b)

= a · (a+b) + b · (a+b)

= a · a + a · b + b · a + b · b

= a² + ab + ba + b²

Since multiplication is commutative i.e., (ab = ba), we can simplify this expression to:

a² + 2ab + b²

∴ Proof demonstrates that (a+b)² is equal to ( a² + 2ab + b²).

Visual Proof,

Proof of (a+b)² = a² + 2ab + b² identity, let’s take a square of side a+b and divide it like the following diagram.

To prove the identity, we have to calculate the area of the square with side (a+b) which is (a+b)².

Identity 2: (a-b)² = a²+b²-2ab

To prove the polynomial identity (a – b)² = a² + b² – 2ab, we can use the distributive property and perform the necessary algebraic steps:

Starting with the left-hand side:

(a – b)² = (a – b) · (a – b)

Using the distributive property:

(a – b)² = a · (a – b) – b · (a – b)

Further simplifying:

(a – b)² = a² – ab – ab + b²

Combining like terms:

(a – b)² = a² – 2ab + b²

Now, the result matches the right-hand side of the given identity:

(a – b)² = a² + b² – 2ab

∴ Polynomial identity (a – b)² = a² + b² – 2ab is proved.

Visual Proof

Proof of (a-b)² = a²-2ab+b² identity, let’s again consider a square but this time with side “a”.

Now, take a small segment “b” from its side and divide the square as follows:

To prove the identity, we have to calculate the area of the square with side (a-b) which is (a-b)².

Identity 3: (a+b)(a-b)² = a²-b²

Use the distributive property (FOIL method) to multiply the two binomials:

(a + b)(a −b)=a (a−b) + b (a−b)

Now, multiply each term separately:

a (a −b) = a²−ab

b (a −b) = ab−b²

Combine the two results:

(a+b)(a–b)=a²−ab+ab−b²

Solving like terms:

(a+b)(a–b)=a²−b²

Hence,the polynomial identity (a+b)(a–b)=a²−b² is proved.

Identity 4: (x + a)(x + b) = x² + x(a + b) + ab

Certainly, let’s prove the polynomial identity (x + a)(x + b) = x² + x(a + b) + ab

Start with the left side of the identity: (x + a)(x + b)

Use the distributive property (FOIL method) to multiply the two binomials:

(x + a)(x + b) = x(x + b) + a(x + b)

Now, multiply each term separately:

x(x + b) = x² + xb

a(x + b) = ax + ab

Combine the two results:

(x + a)(x + b) = x² + xb + ax + a

Combine like terms:

(x + a)(x + b) = x² + x(a + b) + ab

Hence proved the polynomial identity (x + a)(x + b) = x² + x(a + b) + ab

Polynomial identities are mathematical expressions or equations that are true for all values of the variables involved. These identities are particularly useful in simplifying expressions and solving equations involving polynomials.

These are the equations involving polynomials that hold true for all values of the variables involved. These identities are very useful in simplifying expressions and solving equations more efficiently.

It is an equation that hold for all values of the variables within them. These identities are often used to simplify expressions and solve polynomial equations more easily.

It is an equation that holds for all possible values of the variables involved. It establishes a relation between two or more polynomial expressions, regardless of the specific numerical values of the variables. One common example is the polynomial identity (a+b)²=a²+ 2ab +b², which remains true for any values of a and b.

Let’s know more about various identities of polynomials, types of polynomial identities, and their proof along with some solved examples for clear understanding.