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Chapter 5: Problem 16

Factor out the greatest common factor in each expression. $$10 x^{2}-20 y^{3}$$

Short Answer

Expert verified
10(x^2 - 2y^3)

Step by step solution

01

- Identify the coefficients

Look at the coefficients of each term in the expression: 10 and -20. Find the greatest common factor (GCF) of these numbers.
02

- Find the GCF of the coefficients

The GCF of 10 and 20 is 10.
03

- Identify the variables

Look at the variables in each term: \(x^2\) and \(y^3\). Since these terms have different variables, there is no common variable factor.
04

- Factor out the GCF

Factor the GCF from the expression. The GCF is 10, so divide each term by 10 and rewrite the expression. \(10x^2 - 20y^3 = 10(x^2 - 2y^3)\)

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Key Concepts

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

greatest common factor (GCF)
Understanding the Greatest Common Factor (GCF) is essential in simplifying expressions. The GCF of two or more numbers is the largest number that divides all of them without leaving a remainder. In the expression given \(10x^2 - 20y^3\), the coefficients are 10 and -20. To find the GCF, we take the highest number that evenly divides both. Here, 10 is the largest number that divides both 10 and 20, so the GCF is 10. This helps in factoring expressions efficiently, making them simpler to work with.
coefficients
Coefficients are the numerical parts of algebraic terms. In \(10x^2 - 20y^3\), the coefficients are the numbers 10 and -20 attached to the variable terms. Identifying and understanding coefficients is crucial in the factoring process.
To find the GCF of the coefficients, we look for the largest number that can divide each coefficient evenly. For 10 and -20, this number is 10, which is why we can factor out 10 from the expression. This step simplifies complex expressions, making calculations and further operations more manageable.
variable terms
Variable terms are the parts of an algebraic expression that contain variables raised to a power, like \(x^2\) and \(y^3\) in our example. Each term consists of a coefficient and a variable part.
In the given expression, the term \(10x^2\) has the variable part \(x^2\), and the term \(-20y^3\) has the variable part \(y^3\). When factoring, variable terms play a crucial role in determining what can be factored out. In this example, since there are no common variable parts between \(x^2\) and \(y^3\), we only factor out the GCF of the coefficients.
factoring process
The factoring process involves breaking down an expression into simpler parts. In our example, \(10x^2 - 20y^3\), the steps are straightforward.
First, identify the coefficients and variables involved. Next, find the GCF of the coefficients, which is 10. Then, rewrite the expression by factoring out this GCF.
When we divide each term by 10, we get: \(10(x^2 - 2y^3)\). This makes the expression simpler and easier to handle in further calculations or equations.

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Most popular questions from this chapter

Firing a slingshot. A girl uses a slingshot to propel a stone upward at 64 feet per second from a window that is 80 feet above the ground. The height of the stone above the ground (in feet) at time \(t\) (in seconds) is given by $$h(t)=-16 t^{2}+64 t+80$$ Find the time at which the stone strikes the ground.

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