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− b2is 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− b2is 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.
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|1.||What is a PolynomialIdentity?|
|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 identityrefers toan 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 ofx = 4in both cases. If you write the equations in the form ax– b = c, you will see that the two equations are:
ax– b = c2ax – 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 polynomialidentities in math. The four most important polynomialidentities orformulasare 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
Apart from these simple polynomialidentities listed above, there are otheridentities of polynomials. Here are some most commonly used identitiesof 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 identityproofs 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 rectanglewhose sides are x + aand x + b.
The area of arectangle with sidesx + aand x + bin terms of the individual areas of the rectangles and the squareis 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)2is nothing but (a + b) × (a + b). This can be visualizedas a squarewhose sides are (a + b) and areais (a + b)2.
The areaof the square (a + b)2 in terms of the productis (a+b)(a+b). The area of the square (a + b)2is also equal to thesum 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).
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 beginwith a large square of areaa2. We will reduce the length of all sides by b. We now have to remove the extra bits from a2to be left with (a − b)2. In the figure below, (a − b)2 is shown by the blue area.
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 abtwice will alsoremovethe overlapping square at the bottom right cornertwice. Hence, we add b2. Therefore, (a − b)2 = a2− 2ab + b2
Related Articles on Polynomial Identity
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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.
Ifx + 1/x = 6, find the value ofx2+ 1/x2Find the value ofa−b, if (a+b) = 5andab = 4The length and breadth of a rectangle measure2x + 3 units and2x − 3units. Find the area of the rectangle in terms ofx.