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Chapter 6: Problem 24

Factor completely. $$ y^{3}-z^{3} $$

Short Answer

Expert verified
y^{3} - z^{3} = (y - z)(y^{2} + yz + z^{2})

Step by step solution

01

Identify the expression

The given expression is a difference of cubes: y^{3} - z^{3}
02

Recall the difference of cubes formula

The difference of cubes formula is: a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})
03

Apply the formula to the expression

Identify 'a' and 'b' in the expression. Here, 'a' is 'y' and 'b' is 'z'. Substitute these into the formula:y^{3} - z^{3} = (y - z)(y^{2} + yz + z^{2})
04

Write the final factored form

The factored form of the given expression is:y^{3} - z^{3} = (y - z)(y^{2} + yz + z^{2})

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Difference of Cubes
In algebra, the difference of cubes is a specific type of polynomial that we can factor using a special formula. It involves two cubes being subtracted from each other. In mathematical notation, it looks like this: \(a^3 - b^3\). The 'difference' part comes from the subtraction symbol (-).
To break it down, a cube is any number raised to the power of three. So, if you have two cubes, you are dealing with numbers like \(y^3\) and \(z^3\).
In our exercise, we need to factor the expression \(y^3 - z^3\). Recognizing it as a difference of cubes helps us use a straightforward formula to simplify it.
Factoring Formulas
Factoring polynomials can sometimes be tricky, but knowing specific formulas can make it much easier. For the difference of cubes, the formula goes like this: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\). This equation is worth memorizing because it can save you time on your homework.
To apply this formula, first identify what 'a' and 'b' represent in your original polynomial. In our example, \(a = y\) and \(b = z\).
By substituting these values into our formula, we get:
\(y^3 - z^3 = (y - z)(y^2 + yz + z^2)\)
This gives us the expression in its completely factored form!
Algebraic Expressions
An algebraic expression is a mathematical phrase that can contain numbers, variables, and operators (like add or subtract).
For instance, \(y^3 - z^3\) is an algebraic expression involving the variables 'y' and 'z' raised to the third power.
By understanding how to handle these expressions, like recognizing patterns such as the difference of cubes, you can simplify complex problems into more manageable pieces.
Always remember: The goal of factoring is to break down the original expression into products of simpler expressions, making it easier to work with in future calculations or solve equations.
Thus, learning various factoring formulas is crucial for anyone studying algebra.

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Most popular questions from this chapter

During the first 13 sec of a jump, a skydiver falls approximately \(11.12 t^{2}\) feet in \(t\) seconds. A small heavy object (with less wind resistance) falls about \(15.4 t^{2}\) feet in \(t\) seconds. Suppose that a skydiver jumps from \(30,000 \mathrm{ft}\), and 1 sec later a camera falls out of the airplane. How long will it take the camera to catch up to the skydiver?

Factor completely. Assume that variables in exponents represent positive integers. $$ x^{8}-2^{8} $$

Solve. Round any irrational solutions to the nearest thousandth. $$ x^{2}-3 x-7=0 $$

Let \(g(x)=x^{2} .\) Find \(a\) such that \(g(a)=144\).

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ (a+b)^{2}-9 $$
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