If you have ever come across the algebraic expression (a^3 – b^3), you might wonder how to solve it efficiently without expanding the whole expression. Well, you’re in luck! There’s a simple formula known as the difference of cubes formula that can help you with this.
The difference of cubes formula states that (a^3 – b^3) can be factored as ((a – b)(a^2 + ab + b^2)). This formula is derived from the polynomial (a^3 – b^3), which can be factored by observing patterns in the terms.
Let’s delve into the breakdown of this formula and understand how it works step by step.
In the expression (a^3 – b^3), identify the values of (a) and (b).
Substitute the values of (a) and (b) into the formula ((a – b)(a^2 + ab + b^2)). This will give you the factored form of the expression.
Simplify the expression further if possible by combining like terms or performing any necessary operations.
Let’s work through an example problem to illustrate the application of the difference of cubes formula.
Given: (8^3 – 2^3)
In this case, (a = 8) and (b = 2).
Substitute the values into ((a – b)(a^2 + ab + b^2)) to get:
((8 – 2)(8^2 + 8*2 + 2^2))
Calculate the values to get:
(6(64 + 16 + 4))
(6*84)
(504)
Therefore, (8^3 – 2^3 = 504).
The difference of cubes formula finds applications in various fields and equations, including:
Yes, the formula works for both positive and negative values of (a) and (b).
Yes, there are formulas for sum of cubes ((a^3 + b^3)), as well as higher powers like the sum of fourth powers, but the difference of cubes formula is a particularly common and useful one.
One mnemonic to remember the formula is “Square, Opposite, Square”, where you square the first term, change the sign, and then square the last term.
The formula is often used in calculations involving volume, such as in engineering and physics, where cubic terms are common.
While the formula specifically addresses cubes, similar patterns can be observed and used to derive formulas for differences of higher powers.
Now that you have a solid understanding of the difference of cubes formula, you can confidently tackle problems involving cubic terms and factor them efficiently. Practice applying the formula on different examples to strengthen your algebraic skills and problem-solving abilities.
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