%5E3-formula-questions.jpeg "(a+b)^3 Formula Questions")
(a+b)^3 Formula and Practice Questions
The (a+b)^3 formula, also known as the cube of a binomial, is a fundamental concept in algebra. It is used to expand the cube of a sum of two terms. In this article, we will discuss the formula, its derivation, and provide some practice questions to help you master this concept.
The (a+b)^3 Formula
The (a+b)^3 formula is given by:
(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
This formula can be derived by using the distributive property of multiplication over addition and the fact that (a+b)^2 = a^2 + 2ab + b^2.
Derivation of the (a+b)^3 Formula
To derive the (a+b)^3 formula, we can start with the following expression:
(a+b)^3 = (a+b) × (a+b)^2
Using the distributive property, we can expand the right-hand side of the equation as follows:
(a+b) × (a^2 + 2ab + b^2)
= a(a^2 + 2ab + b^2) + b(a^2 + 2ab + b^2)
= a^3 + 2a^2b + ab^2 + ba^2 + 2ab^2 + b^3
= a^3 + 3a^2b + 3ab^2 + b^3
Thus, we have derived the (a+b)^3 formula.
Practice Questions
Here are some practice questions to help you master the (a+b)^3 formula:
Question 1
Expand (x+2)^3 using the (a+b)^3 formula.
Solution
Using the formula, we get:
(x+2)^3 = x^3 + 3x^2(2) + 3x(2)^2 + 2^3
= x^3 + 6x^2 + 12x + 8
Question 2
Expand (2y-3)^3 using the (a+b)^3 formula.
Solution
Using the formula, we get:
(2y-3)^3 = (2y)^3 + 3(2y)^2(-3) + 3(2y)(-3)^2 + (-3)^3
= 8y^3 - 18y^2 - 27y + 27
Question 3
Expand (a+4)^3 using the (a+b)^3 formula.
Solution
Using the formula, we get:
(a+4)^3 = a^3 + 3a^2(4) + 3a(4)^2 + 4^3
= a^3 + 12a^2 + 48a + 64
I hope these practice questions help you understand and apply the (a+b)^3 formula. Remember to practice regularly to master this concept!
