Lesson 5 Homework Practice Factoring Linear Expressions Answers

4 views

Skip to first unread message

Torrie Dimaggio

unread,

Dec 10, 2023, 3:34:25 AM12/10/23

to PhyML forum

Lesson 5 Homework Practice Factoring Linear Expressions Answers

Factoring linear expressions is a skill that can help you simplify algebraic expressions and solve equations. In this article, you will learn how to factor linear expressions by finding the greatest common factor (GCF) of the terms and rewriting the expression as a product. You will also find some practice problems and answers to help you master this topic.

lesson 5 homework practice factoring linear expressions answers

Download File https://jfilte.com/2wJyQg

What is a linear expression?

A linear expression is an algebraic expression that consists of one or more terms that have a variable raised to the first power. For example, 3x + 5 and 2x - 7y + 4 are linear expressions. A term is a part of an expression that is separated by a plus or minus sign. For example, 3x and 5 are terms in the expression 3x + 5.

What is factoring?

Factoring is a process of breaking down an expression into smaller parts that can be multiplied together to get the original expression. For example, 6x + 9 can be factored into 3(2x + 3), because 3(2x + 3) = 6x + 9. The smaller parts are called factors, and the original expression is called the product.

How to factor linear expressions?

To factor linear expressions, you need to find the GCF of the terms and divide each term by the GCF. Then, you can write the expression as a product of the GCF and another expression that contains the quotients. For example, to factor 4x + 12, you need to find the GCF of 4x and 12, which is 4. Then, you can divide each term by 4 and get x + 3. Finally, you can write the expression as a product of 4 and x + 3: 4x + 12 = 4(x + 3).

Lesson 5 Homework Practice Factoring Linear Expressions

Here are some practice problems for you to try. Factor each expression. If the expression cannot be factored, write cannot be factored.

8r - 14

5x + 35

32x - 15

24 + 32x

25x + 14

The answers are:

8r - 14 = 2(4r - 7)

5x + 35 = 5(x + 7)

32x - 15 = cannot be factored

24 + 32x = 8(3 + 4x)

25x + 14 = cannot be factored

We hope this article helped you understand how to factor linear expressions and practice this skill. For more resources on algebraic expressions, check out our other articles on this topic.

Why is factoring linear expressions important?

Factoring linear expressions is important because it can help you simplify expressions and solve equations. For example, if you have an equation like 4x + 12 = 0, you can factor the expression 4x + 12 into 4(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 factors is zero, then one or both of the factors must be zero. So, if 4(x + 3) = 0, then either 4 = 0 or x + 3 = 0. Since 4 cannot be zero, we can conclude that x + 3 = 0, and then solve for x by subtracting 3 from both sides: x = -3. Therefore, the solution of the equation is x = -3.

How to check your answers when factoring linear expressions?

When you factor a linear expression, you can check your answer by using the distributive property to multiply the factors and see if you get back the original expression. The distributive property states that a(b + c) = ab + ac. For example, if you factor the expression 6x - 9 into 3(2x - 3), you can check your answer by multiplying 3 and 2x - 3 using the distributive property: 3(2x - 3) = 3(2x) + 3(-3) = 6x - 9. Since this is the same as the original expression, you can be confident that your answer is correct.

How to use algebra tiles to factor linear expressions?

Algebra tiles are a visual tool that can help you understand and manipulate algebraic expressions. Algebra tiles are rectangular or square shapes that represent different terms in an expression. For example, a small square tile represents 1, a long rectangle tile represents x, and a large square tile represents x^2. You can use algebra tiles to model linear expressions by arranging them into a rectangle shape. For example, to model the expression 4x + 12, you can use four long rectangle tiles and twelve small square tiles to form a rectangle that is four tiles long and three tiles wide.

To factor linear expressions using algebra tiles, you need to find the dimensions of the rectangle that models the expression. The length and width of the rectangle are the factors of the expression. For example, to factor the expression 4x + 12, you can look at the rectangle you formed and see that it has a length of 4 and a width of x + 3. Therefore, 4x + 12 = 4(x + 3).

How to factor linear expressions with three terms?

Factoring linear expressions with three terms is similar to factoring linear expressions with two terms, but you need to find the GCF of all three terms instead of just two. For example, to factor the expression 6x - 9 + 15y, you need to find the GCF of 6x, -9, and 15y, which is 3. Then, you can divide each term by 3 and get 2x - 3 + 5y. Finally, you can write the expression as a product of 3 and another expression that contains the quotients: 6x - 9 + 15y = 3(2x - 3 + 5y).

How to factor linear expressions with fractions?

Factoring linear expressions with fractions is similar to factoring linear expressions with whole numbers, but you need to find the GCF of the numerators and the denominators separately. For example, to factor the expression 6/9x + 12/9, you need to find the GCF of 6 and 12, which is 6, and the GCF of 9 and 9, which is 9. Then, you can divide each term by 6/9 and get x + 2. Finally, you can write the expression as a product of 6/9 and another expression that contains the quotients: 6/9x + 12/9 = (6/9)(x + 2).

How to factor linear expressions with negative coefficients?

Factoring linear expressions with negative coefficients is similar to factoring linear expressions with positive coefficients, but you need to be careful with the signs. For example, to factor the expression -4x - 12, you need to find the GCF of -4x and -12, which is -4. Then, you can divide each term by -4 and get x + 3. Finally, you can write the expression as a product of -4 and another expression that contains the quotients: -4x - 12 = -4(x + 3). Note that when you divide a negative term by a negative factor, you get a positive quotient.

How to factor linear expressions with variables in the denominator?

Factoring linear expressions with variables in the denominator is similar to factoring linear expressions with whole numbers in the denominator, but you need to find the GCF of the numerators and the denominators separately. For example, to factor the expression 6/x + 12/x^2, you need to find the GCF of 6 and 12, which is 6, and the GCF of x and x^2, which is x. Then, you can divide each term by 6/x and get 1 + 2/x. Finally, you can write the expression as a product of 6/x and another expression that contains the quotients: 6/x + 12/x^2 = (6/x)(1 + 2/x).

How to factor linear expressions with parentheses?

Factoring linear expressions with parentheses is similar to factoring linear expressions without parentheses, but you need to apply the distributive property first to remove the parentheses. The distributive property states that a(b + c) = ab + ac. For example, to factor the expression 3(2x - 5) + 9, you need to distribute 3 to each term inside the parentheses and get 6x - 15 + 9. Then, you can factor this expression as usual by finding the GCF of the terms and dividing each term by the GCF. The GCF of 6x, -15, and 9 is 3. So, you can divide each term by 3 and get 2x - 5 + 3. Finally, you can write the expression as a product of 3 and another expression that contains the quotients: 3(2x - 5) + 9 = 3(2x - 5 + 3).

How to factor linear expressions with exponents?

Factoring linear expressions with exponents is similar to factoring linear expressions without exponents, but you need to apply the power of a product rule first to remove the exponents. The power of a product rule states that (ab)^n = a^n b^n. For example, to factor the expression (2x)^3 + 12x^2, you need to apply the power of a product rule to the first term and get 8x^3 + 12x^2. Then, you can factor this expression as usual by finding the GCF of the terms and dividing each term by the GCF. The GCF of 8x^3 and 12x^2 is 4x^2. So, you can divide each term by 4x^2 and get 2x + 3. Finally, you can write the expression as a product of 4x^2 and another expression that contains the quotients: (2x)^3 + 12x^2 = 4x^2(2x + 3).

How to factor linear expressions with different variables?

Factoring linear expressions with different variables is similar to factoring linear expressions with one variable, but you need to find the GCF of each variable separately. For example, to factor the expression 6xy + 9xz, you need to find the GCF of 6 and 9, which is 3, the GCF of x and x, which is x, and the GCF of y and z, which is 1. Then, you can multiply these factors together and get 3x as the GCF of the expression. Then, you can divide each term by 3x and get 2y + 3z. Finally, you can write the expression as a product of 3x and another expression that contains the quotients: 6xy + 9xz = 3x(2y + 3z).

How to factor linear expressions with coefficients that are not whole numbers?

Factoring linear expressions with coefficients that are not whole numbers is similar to factoring linear expressions with coefficients that are whole numbers, but you need to find the GCF of the coefficients as fractions or decimals. For example, to factor the expression 0.5x + 1.5, you need to find the GCF of 0.5 and 1.5, which is 0.5. Then, you can divide each term by 0.5 and get x + 3. Finally, you can write the expression as a product of 0.5 and another expression that contains the quotients: 0.5x + 1.5 = 0.5(x + 3).

How to factor linear expressions with more than one variable?

Factoring linear expressions with more than one variable is similar to factoring linear expressions with one variable, but you need to find the GCF of each variable separately and multiply them together. For example, to factor the expression 6xy + 9xz + 12yz, you need to find the GCF of 6, 9, and 12, which is 3, the GCF of x and x, which is x, the GCF of y and y, which is y, and the GCF of z and z, which is z. Then, you can multiply these factors together and get 3xyz as the GCF of the expression. Then, you can divide each term by 3xyz and get 2 + 3x + 4y. Finally, you can write the expression as a product of 3xyz and another expression that contains the quotients: 6xy + 9xz + 12yz = 3xyz(2 + 3x + 4y).

Conclusion

Factoring linear expressions is a useful skill that can help you simplify expressions and solve equations. In this article, you learned how to factor linear expressions by finding the greatest common factor of the terms and rewriting the expression as a product. You also learned how to factor linear expressions with different types of coefficients, variables, exponents, fractions, parentheses, and more. You also practiced factoring linear expressions with some examples and exercises. We hope this article helped you understand how to factor linear expressions and apply this skill to various problems. For more resources on algebraic expressions, check out our other articles on this topic.