How to Factor Linear Expressions: A Guide for Lesson 5 Homework
Factoring linear expressions is a skill that can help you simplify algebraic equations and solve them more easily. In this article, we will explain what factoring linear expressions means, how to do it step by step, and provide some examples and answers from lesson 5 homework.
What is Factoring Linear Expressions
Factoring linear expressions means finding two or more terms that can be multiplied together to get the original expression. For example, the expression 2x + 6 can be factored into 2(x + 3), because 2 times (x + 3) equals 2x + 6. Factoring linear expressions can make them easier to work with, especially when solving equations or finding common factors.
How to Factor Linear Expressions: Step by Step
To factor linear expressions, you need to follow these steps:
Identify the greatest common factor (GCF) of all the terms in the expression. The GCF is the largest number or variable that can divide evenly into all the terms. For example, the GCF of 2x + 6 is 2, because 2 can divide into both 2x and 6.
Divide each term by the GCF and write the result in parentheses. For example, dividing 2x + 6 by 2 gives (x + 3).
Write the GCF outside the parentheses and multiply it by the expression inside. For example, writing 2 outside the parentheses and multiplying it by (x + 3) gives 2(x + 3), which is the factored form of 2x + 6.
Examples and Answers from Lesson 5 Homework
Here are some examples and answers from lesson 5 homework on factoring linear expressions. Try to factor them on your own first, then check your answers below.
Example: Factor 3y - 9
Answer: The GCF of 3y - 9 is 3, so we divide each term by 3 and get (y - 3). Then we write 3 outside the parentheses and multiply it by (y - 3) to get the final answer: 3(y - 3).
Example: Factor x^2 + x - 6
Answer: The GCF of x^2 + x - 6 is 1, so we cannot factor out any number or variable. However, we can still factor this expression by finding two numbers that add up to the coefficient of x (which is 1) and multiply to the constant term (which is -6). In this case, those numbers are -2 and 3, because -2 + 3 = 1 and -2 times 3 = -6. So we write (x - 2) and (x + 3) as the factors of x^2 + x - 6. Then we write them next to each other to get the final answer: (x - 2)(x + 3).
Example: Factor -4a^2 + a - b
Answer: The GCF of -4a^2 + a - b is -1, so we divide each term by -1 and get (4a^2 - a + b). Then we write -1 outside the parentheses and multiply it by (4a^2 - a + b) to get the final answer: -1(4a^2 - a + b).
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Why is Factoring Linear Expressions Important
Factoring linear expressions is important because it can help you solve equations and inequalities that involve linear expressions. For example, if you want to solve the equation 2x + 6 = 0, you can factor the left side into 2(x + 3) and then use the zero product property to find the value of x that makes the equation true. The zero product property states that if a product of two or more factors is equal to zero, then at least one of the factors must be zero. So, in this case, we have 2(x + 3) = 0, which means either 2 = 0 or x + 3 = 0. Since 2 cannot be zero, we only need to solve x + 3 = 0, which gives x = -3 as the solution.
Factoring linear expressions can also help you find common factors and simplify fractions that involve linear expressions. For example, if you want to simplify the fraction (2x + 6) / (4x + 12), you can factor both the numerator and the denominator into 2(x + 3) and then cancel out the common factor of 2. This gives you (x + 3) / (2x + 6) as the simplified fraction.
How to Practice Factoring Linear Expressions
The best way to practice factoring linear expressions is to do lots of exercises and check your answers with a reliable source. You can find many online resources that offer practice problems and solutions on factoring linear expressions, such as Khan Academy, Math Planet, Math Is Fun, and more. You can also use a calculator or a software program that can factor linear expressions for you, such as Wolfram Alpha, Symbolab, Mathway, and more. However, you should not rely on these tools too much and try to understand the steps and logic behind factoring linear expressions.
Another way to practice factoring linear expressions is to create your own problems and challenge yourself or your friends. You can start with simple expressions that have small numbers and coefficients and then gradually increase the difficulty by using larger numbers and coefficients, negative signs, fractions, variables with higher powers, and more. You can also mix up different types of factoring methods, such as factoring by grouping, factoring by difference of squares, factoring by sum or difference of cubes, and more. The more you practice factoring linear expressions, the more confident and proficient you will become.
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