Factor $20r + 60t$ To Identify The Equivalent Expressions.Choose 2 Answers:A. $4(5r + 15t)$ B. $ 5 ( 5 R + 12 ) 5(5r + 12) 5 ( 5 R + 12 ) [/tex] C. $60(20r + T)$ D. 20 ( R + 3 T ) 20(r + 3t) 20 ( R + 3 T ) $
Introduction
Algebraic expressions are a fundamental concept in mathematics, and factoring them is a crucial skill to master. Factoring involves expressing an algebraic expression as a product of simpler expressions, known as factors. In this article, we will focus on factoring the expression $20r + 60t$ and identify the equivalent expressions.
Understanding the Expression
Before we dive into factoring, let's understand the given expression. The expression $20r + 60t$ is a linear combination of two variables, r and t. The coefficients of r and t are 20 and 60, respectively.
Factoring the Expression
To factor the expression $20r + 60t$, we need to find the greatest common factor (GCF) of the coefficients 20 and 60. The GCF of 20 and 60 is 20.
# Factoring the Expression
Step 1: Identify the Greatest Common Factor (GCF)
 "Step 1: Identify the Greatest Common Factor (GCF)")
The GCF of 20 and 60 is 20.
Step 2: Factor out the GCF
We can factor out the GCF from both terms:
Analyzing the Options
Now that we have factored the expression, let's analyze the options:
A. $4(5r + 15t)$
B. $5(5r + 12)$
C. $60(20r + t)$
D. $20(r + 3t)$
Option A: $4(5r + 15t)$
Option A is not correct because the GCF of 20 and 60 is 20, not 4.
Option B: $5(5r + 12)$
Option B is not correct because the GCF of 20 and 60 is 20, not 5, and the constant term is not 12.
Option C: $60(20r + t)$
Option C is not correct because the GCF of 20 and 60 is 20, not 60.
Option D: $20(r + 3t)$
Option D is correct because the GCF of 20 and 60 is 20, and the expression is factored correctly.
Conclusion
In conclusion, the correct factorization of the expression $20r + 60t$ is $20(r + 3t)$. This is achieved by identifying the GCF of the coefficients and factoring it out from both terms.
Tips and Tricks
- Always identify the GCF of the coefficients before factoring.
- Factor out the GCF from both terms.
- Check your work by multiplying the factors together to ensure that you get the original expression.
Practice Problems
Try factoring the following expressions:
Answer Key
Introduction
In our previous article, we discussed how to factor the expression $20r + 60t$ and identified the equivalent expressions. In this article, we will provide a Q&A guide to help you understand the concept of factoring algebraic expressions.
Q: What is factoring in algebra?
A: Factoring involves expressing an algebraic expression as a product of simpler expressions, known as factors. This is achieved by identifying the greatest common factor (GCF) of the coefficients and factoring it out from both terms.
Q: How do I identify the greatest common factor (GCF)?
A: To identify the GCF, you need to find the largest number that divides both coefficients without leaving a remainder. For example, the GCF of 20 and 60 is 20.
Q: What is the difference between factoring and simplifying?
A: Factoring involves expressing an algebraic expression as a product of simpler expressions, while simplifying involves combining like terms to reduce the complexity of the expression.
Q: Can I factor an expression with a negative coefficient?
A: Yes, you can factor an expression with a negative coefficient. For example, the expression $-3x + 9y$ can be factored as $-3(x - 3y)$.
Q: How do I factor an expression with a variable in the coefficient?
A: To factor an expression with a variable in the coefficient, you need to identify the GCF of the coefficients and factor it out from both terms. For example, the expression $2x^2 + 6x$ can be factored as $2x(x + 3)$.
Q: Can I factor an expression with a fraction?
A: Yes, you can factor an expression with a fraction. For example, the expression $\frac{2}{3}x + \frac{4}{3}y$ can be factored as $\frac{2}{3}(x + 2y)$.
Q: How do I factor an expression with a binomial?
A: To factor an expression with a binomial, you need to identify the GCF of the coefficients and factor it out from both terms. For example, the expression $x^2 + 5x$ can be factored as $x(x + 5)$.
Q: Can I factor an expression with a trinomial?
A: Yes, you can factor an expression with a trinomial. For example, the expression $x^2 + 7x + 12$ can be factored as $(x + 3)(x + 4)$.
Q: How do I factor an expression with a quadratic?
A: To factor an expression with a quadratic, you need to identify the GCF of the coefficients and factor it out from both terms. For example, the expression $x^2 + 4x + 4$ can be factored as $(x + 2)^2$.
Conclusion
In conclusion, factoring algebraic expressions is a crucial skill to master in mathematics. By understanding the concept of factoring and practicing with different types of expressions, you will become proficient in factoring and be able to solve a wide range of problems.
Tips and Tricks
- Always identify the GCF of the coefficients before factoring.
- Factor out the GCF from both terms.
- Check your work by multiplying the factors together to ensure that you get the original expression.
Practice Problems
Try factoring the following expressions:
Answer Key
By following the steps outlined in this article, you should be able to factor algebraic expressions with ease. Remember to always identify the GCF of the coefficients and factor it out from both terms. With practice, you will become proficient in factoring algebraic expressions and be able to solve a wide range of problems.