Proof That Says All Polynomials Are Continuous
In Mathematics, a polynomial identity is an equation that helps us to easily solve expressions involving larger exponents and numbers. Generally, polynomial identities are used for the expansion of the expression, by breaking down the expression into simpler units. In this article, we will learn the definition of polynomial identity, examples and proofs for some of the important polynomial identities in detail.
Table of Contents:
- Definition
- Important Polynomial Identities
- Proof
- Identity 1
- Identity 2
- Identity 3
- Identity 4
- Examples
- FAQs
What is Polynomial Identity?
Polynomial identity refers to an equation that is always true, irrespective of the variable values. We use polynomial identities while factorising the polynomial or expanding the polynomial. To solve the polynomial equation easily, it is essential to learn some of the important polynomial identities which are listed below.
Important Polynomial Identities
The most important polynomial identities, also known as algebraic identities in Maths are:
- (a+b)2 = a2+b2+2ab
- (a-b)2 = a2+b2-2ab
- (a+b)(a-b) = a2-b2
- (x+a)(x+b) = x2+ x(a+b)+ab
Apart from the above-mentioned identities, there are other polynomial identities that are equally important while solving the expressions. They are:
- (a+b)3 = a3+3a2b+3ab2+b3
- (a-b)3 = a3-3a2b+3ab2-b3
- a3+b3 = (a+b)(a2-ab+b2)
- a3-b3 = (a-b)(a2+ab+b2)
- (a+b+c)2 = a2+b2+c2+2ab+2bc+2ca
Proof of Polynomial Identities
Here, we are going to discuss the proofs of the above-mentioned polynomial identities, one by one.
Identity 1: (a+b)2 = a2+b2+2ab
Here, (a+b)2 is nothing but the product of (a+b) and (a+b).
(i.e) (a+b)2 = (a+b)×(a+b)
So, this can be visualised as a square whose side is a+b and its area is given by (a+b)2.
From the figure, we can say that the area of a square (a+b)2 is equal to the sum of the individual squares and rectangles.
Hence, (a+b)2 = a2+b2+2ab is proved.
Identity 2: (a-b)2 = a2+b2-2ab
Let us assume that (a-b)2 is the area of a square of length (a-b). To understand this, let us start with the larger square of area "a2". Reduce all the sides of a square by length "b". Now, remove the extra bits from a2 and we are left with (a-b)2, which is represented by the yellow area.
So, to get the yellow area from the larger blue square, subtract the horizontal and vertical strips which have the area ab. However, removing ab twice will also eliminate the overlapping square at the bottom right side corner twice. So, we add b2.
Hence, (a-b)2 = a2+b2-2ab is proved.
Identity 3: (a+b)(a-b) = a2-b2
Assume that (a+b)(a-b) is the area of a rectangle whose sides are (a+b) and (a-b).
Thus, by rearranging the individual squares and rectangles, we will get (a+b)(a-b) = a2-b2.
Hence, proved.
Identity 4: (x+a)(x+b) = x2+x(a+b)+ab
Let us assume that (x+a)(x+b) is the area of a rectangle whose sides are (x+a) and (x+b).
Thus, the area of a rectangle in terms of individual squares and rectangles is x2+ax+bx+b2.
Hence, (x+a)(x+b) = x2+x(a+b)+b2 is proved.
Examples
Go through the problems given below to understand how to solve expressions by applying polynomial identities.
Example 1:
Find the side length of a square whose area is 9x2+4+12x.
Solution:
Given that the area of a square is 9x2+4+12x, which is equal to 9x2+22+12x.
The given polynomial resembles the identity a2+b2+2ab.
Hence, a2 = 9x2
⇒a = 3x
b2 = 22
⇒b =2
2ab = 12x = 2×2×3x
Thus, (a+b)2 = (3x+2)2, which is the area of a square.
Hence, the side length of a square is (3x+2).
Example 2:
Find (5x+3)2 using the polynomial identity.
Solution:
Given expression: (5x+3)2.
We know that (a+b)2 = a2+b2+2ab.
Here, a = 5x, b = 3.
Therefore, (5x+3)2 = (5x)2+ (3)2+ 2(5x)(3)
(5x+3)2 = 25x2+9+30x.
Hence, (5x+3)2 = 25x2+30x+9.
Example 3:
Simplify the expression: (2a+3)(2a-3).
Solution:
Given expression: (2a+3)(2a-3)
The given expression is similar to (a+b)(a-b).
As we know, (a+b)(a-b) = a2-b2.
Therefore, (2a+3)(2a-3) = (2a)2-(3)3
(2a+3)(2a-3) = 4a2-9.
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Frequently Asked Questions on Polynomial Identity
What is meant by polynomial identity?
Polynomial identity is defined as an equation that is always true regardless of variable values.
When can we use polynomial identities?
We can use polynomial identities while factorising or expanding a polynomial.
What are four important polynomial identities?
The four important polynomial identities are:
(a+b)2 = a2+b2+2ab
(a-b)2 = a2+b2-2ab
(a+b)(a-b) = a2-b2
(x+a)(x+b) = x2+x(a+b)+ab
What is the coefficient of xy in the polynomial (4x+2y)2?
The coefficient of xy in the polynomial (4x+2y)2 is 16.
(i.e.) = (4x+2y)2 = 16x2+4y2+16xy.
Is a(a+5) = a2+5a an identity?
Yes, a(a+5) = a2+5a is an identity.
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