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Polynomial Identity

Polynomial identity is a mathematical fact or equation that helps us to quickly solve expressions involving larger numbers and exponents. It helps in the expansion of an expression by breaking the numbers into simpler units. (a + b)(a − b) = a2 − b2 is a polynomial equation that stands in contrast to a polynomial identity, where both expressions represent the same polynomial in different forms. Consequently, any evaluation of both members gives valid equality. (a + b)(a − b) = a2 − b2 is a polynomial equation that stands in contrast to a polynomial identity, where both expressions represent the same polynomial in different forms. Consequently, any evaluation of both members gives valid equality.

1. What is a Polynomial Identity?
2. Important Polynomial Identities
3. How do you Prove Polynomial Identities?
4. Solved Examples on Polynomial Identity
5. Practice Questions on Polynomial Identity
6. FAQs on Polynomial Identity

What is a Polynomial Identity?

Polynomial identity refers to an equation that is always true regardless of the values assigned to the variables. For the expansion or for the factorization of polynomials, we use polynomial identities.

Polynomial Identity Examples

Consider the equations: 4x - 2 = 14  and  8x - 4 = 28. If you solve both equations separately, you will observe that the value of x = 4 in both cases. If you write the equations in the form ax – b = c, you will see that the two equations are:

  • ax – b = c
  • 2ax – 2b = 2c

Most equations in math work only for certain values.

Example: 4x + 5 = 17  is true only if  x = 3 

Identities are useful because they are always true regardless of the value of the variables.

Important Polynomial Identities

It is very important that we learn about polynomial identities in math. The four most important polynomial identities or formulas are listed below.

Polynomial Identities:  

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

Apart from these simple polynomial identities listed above, there are other identities of polynomials. Here are some most commonly used identities of polynomials:

  • (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
  • (a + b)3 = a3 + 3a2b + 3ab2 + b3
  • (a − b)3 = a3 − 3a2b+ 3ab2 − b3
  • (a)3 + (b)3 = (a + b)(a2 − ab + b2)
  • (a)3 − (b)3 = (a − b)(a2 + ab + b2)
  • (a)3 + (b)3 + (c)3 − 3abc = (a + b + c)(a2 + b2 + c2 − ab − bc−ca)

How do you Prove Polynomial Identities?

In this section, we are going to learn how to prove above mentioned polynomial identities. Some most commonly used polynomial identity proofs are shown below:

Proof of (x + a)(x + b) = x2 + x(a + b) + ab

(x + a)(x + b) is nothing but the area of a rectangle whose sides are x + a and x + b.

Proof of (x + a)(x + b) = x^2 + x(a + b) + ab

The area of a rectangle with sides x + a and x + b in terms of the individual areas of the rectangles and the square is x2 + ax + bx + b2 = x2 + (a + b)x + b2. Therefore, (x + a)(x + b) = x2 + x(a + b) + ab.

Proof of (a + b)2 = a2 + 2ab + b2

(a + b)2 is nothing but (a + b) × (a + b). This can be visualized as a square whose sides are (a + b) and area is (a + b)2.

Proof of (a + b)^2 = a^2 + 2ab + b^2

The area of the square (a + b)2 in terms of the product is (a+b)(a+b). The area of the square (a + b)2 is also equal to the sum of the areas of the individual squares and rectangles. Therefore, (a + b)2 = a2 + 2ab + b2.

Proof of (a + b)(a − b) = a2 − b2

(a + b) (a - b) can be visualized as the area of a rectangle whose sides are (a + b) and (a - b).

Proof of (a + b)(a − b) = a^2 − b^2

Rearranging the individual squares and rectangles, we get: (a + b)(a − b) = a2 − b2.

Proof of (a − b)2 = a2 − 2ab + b2

Once again, let’s think of (a − b)2 as the area of a square with length (a - b). To understand this, let's begin with a large square of area a2. We will reduce the length of all sides by b. We now have to remove the extra bits from a2 to be left with (a − b)2. In the figure below, (a − b)2 is shown by the blue area. 

Proof of (a − b)^2 = a^2 − 2ab + b^2

To get the blue square from the larger orange square, we have to subtract the vertical and horizontal strips that have the area ab. However, removing ab twice will also remove the overlapping square at the bottom right corner twice. Hence, we add b2. Therefore, (a − b)2 = a2 − 2ab + b2

Related Articles on Polynomial Identity

Check out these interesting articles on Polynomial Identity. Click to know more!

Important Notes

  • Remember these four basic identities.

                 (a + b)2 = a2 + 2ab + b2

                 (a − b)2 = a2 − 2ab + b2

                 (a + b)(a − b) = a2 − b2

                (x + a)(x + b) = x2 + x(a + b) + ab

  • While solving problems related to polynomial identities, identify the pattern to check if it has the simplified form or the factored form, and then apply the identity and solve.

Challenging Questions

  • If x + 1/x = 6, find the value of x2 + 1/x2
  • Find the value of a−b, if (a+b) = 5 and ab = 4
  • The length and breadth of a rectangle measure 2x + 3 units and 2x − 3 units. Find the area of the rectangle in terms of x.
 

Solved Examples on Polynomial Identity

  1. Example 1: Using polynomial identities, find (2x - 3y)2.

    Solution:

    Here, we use the identity (a − b)2 = a2 − 2ab + b2 to expand this. Here, a = 2x and b = 3y. Then we get (2x − 3y)2 = (2x)2 − 2(2x)(3y) + (3y)2  = 4x2 − 12xy + 9y2

    Therefore, (2x − 3y)2 = 4x2 − 12xy + 9y2

  2. Example 2: Expand (pqr + 2)(pqr - 5)

    Solution:

    This is based on the identity: (x + a)(x + b) = x2 + x(a + b) + ab, where x = pqr, a = 2, and b = -5. Substituting in the above equation, ( pqr + 2 )( pqr - 5 ) = (pqr)2 + pqr(2 + (−5)) + (2)(−5) =  p2q2r2 − 3pqr − 10

    Therefore, ( pqr + 2 )( pqr - 5 ) = p2q2r2 − 3pqr − 10

  3. Example 3: Help Andrea simplify (7x + 4y)2 + (7x - 4y)2.

    Solution:

    To solve this, we need to use the following identities:

    (a + b)2 = a2 + 2ab + b2

    (a − b)2 = a2 − 2ab + b2

    Here, a = 7x and b = 4y. Substituting the values, we get, (7x + 4y)2 + (7x - 4y)2 = (49x2 + 56xy + 16y2) + (49x2 − 56xy + 16y2) = 98x2 + 32y2

    Therefore, (7x + 4y)2 + (7x - 4y)2 = 98x2 + 32y2

  4. Example 4: The area of a square is 9x2 + 12x + 4. What is the measure of its side?

    Solution:

    This looks like the RHS of the identity: (a + b)2 = a2 + 2ab + b2 where,

    a2 = 9x2 ⇒ a = 3x

    b2 = 4 ⇒ b = 2

    2 × 3x × 2 = 2ab

    So, the area of the square in terms of the product of its sides is the LHS of the identity (a+b)2, which is (3x + 2)2

    Therefore, Side of the Square = (3x + 2)

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FAQs on Polynomial Identity

What is a Polynomial Identity?

Polynomial identity is an equation that is always true regardless of the values assigned to the variables.

How do you Prove Polynomial Identities?

Polynomial identities can be proven by simplifying identity through the application of algebraic identities of polynomials and principles.  

What can Polynomial Identities Apply to Beyond just Polynomials?

Apart from the factorization or expansion of polynomials, polynomial identities can be used to prove or describe numerical relationships.

What are the Identities of Algebra?

Here are some most commonly used formulas of algebra:

  • a2 − b2 = (a + b)(a − b)
  • (a + b)2 = a2 + 2ab + b2
  • (a − b)2 = a2 − 2ab + b2
  • (x + a)(x + b) = x2 + x(a + b) + ab
  • (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
  • (a + b)3 = a3 + 3a2b + 3ab2 + b3
  • (a − b)3 = a3 − 3a2b+ 3ab2 − b3

How do you know if an Equation is an Identity?

An identity is an equation that is always true regardless of the value that is assigned to the variable.

What is an Example of an Equation Identity?

The examples of simple algebraic identities are:

  • (a + b)2 = a2 + 2ab + b2
  • (a − b)2 = a2 − 2ab + b2

What are the Important Polynomial Identities?

The four most important polynomial identities or formulas are listed below.

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

What is the Difference Between an Equation and an Identity?

An equation is a mathematical statement with an "equal to" symbol between two algebraic expressions that have equal values. An identity is an equation that is always true regardless of the value that is assigned to the variable.

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