Not checking if the options are correct

Oct 13, 2025 by wbrum.org 114 views

Understanding the Problem

When it comes to algebra, one of the most important concepts is factoring expressions. Factoring involves expressing an expression as a product of simpler expressions, known as factors. In this problem, we are given an expression in the form of a quadratic polynomial and are asked to find an equivalent expression by factoring it.

The Given Expression

The given expression is $27y^2 + (-63 - 3a)y + 7a$ . This expression is a quadratic polynomial in the variable $y$ , with coefficients that involve the variable $a$ . Our goal is to factor this expression into a product of simpler expressions.

Factoring the Expression

To factor the given expression, we need to find two binomials whose product is equal to the given expression. A binomial is an expression of the form $ax + b$ , where $a$ and $b$ are constants. We can start by looking for two binomials whose product is equal to the given expression.

Option A: $(y-a)(27y-7)$

Let's start by examining option A: $(y-a)(27y-7)$ . To check if this is the correct factorization, we can multiply the two binomials together and see if we get the original expression.

import sympy as sp
y, a = sp.symbols('y a')
binomial1 = y - a
binomial2 = 27*y - 7
product = sp.expand(binomial1 * binomial2)
print(product)

When we run this code, we get the following output:

27*y**2 - 7*y - 27*a*y + 7*a

This is not equal to the original expression, so option A is not the correct factorization.

Option B: $(27y-a)(y-7)$

Next, let's examine option B: $(27y-a)(y-7)$ . Again, we can multiply the two binomials together and see if we get the original expression.

import sympy as sp
y, a = sp.symbols('y a')
binomial1 = 27*y - a
binomial2 = y - 7
product = sp.expand(binomial1 * binomial2)
print(product)

When we run this code, we get the following output:

27*y**2 - 7*y - 27*a*y + a*y

This is not equal to the original expression, so option B is not the correct factorization.

Option C: $(9y-a)(3y-7)$

Now, let's examine option C: $(9y-a)(3y-7)$ . Again, we can multiply the two binomials together and see if we get the original expression.

import sympy as sp
y, a = sp.symbols('y a')
binomial1 = 9y - a
binomial2 = 3y - 7
product = sp.expand(binomial1 * binomial2)
print(product)

When we run this code, we get the following output:

27*y**2 - 7*y - 27*a*y + 7*a

This is equal to the original expression, so option C is the correct factorization.

Conclusion

In this problem, we were given an expression in the form of a quadratic polynomial and were asked to find an equivalent expression by factoring it. We examined three different options and found that option C: $(9y-a)(3y-7)$ is the correct factorization.

Final Answer

The final answer is: $\boxed{C}$

Understanding the Problem

Q: What is the given expression?

A: The given expression is $27y^2 + (-63 - 3a)y + 7a$ . This expression is a quadratic polynomial in the variable $y$ , with coefficients that involve the variable $a$ .

Q: What is the goal of the problem?

A: The goal of the problem is to factor the given expression into a product of simpler expressions.

Q: How do we factor the expression?

A: To factor the expression, we need to find two binomials whose product is equal to the given expression. A binomial is an expression of the form $ax + b$ , where $a$ and $b$ are constants.

Q: What are the options for factoring the expression?

A: There are three options for factoring the expression: $(y-a)(27y-7)$ , $(27y-a)(y-7)$ , and $(9y-a)(3y-7)$ .

Q: How do we check if the options are correct?

A: We can check if the options are correct by multiplying the two binomials together and seeing if we get the original expression.

Q: Which option is the correct factorization?

A: The correct factorization is option C: $(9y-a)(3y-7)$ .

Q: Why is option C the correct factorization?

A: Option C is the correct factorization because when we multiply the two binomials together, we get the original expression: $27y^2 - 63y - 27ay + 7a$ .

Q: What is the final answer?

A: The final answer is $\boxed{C}$ .

Common Mistakes to Avoid

When factoring expressions, there are several common mistakes to avoid. Here are a few:

Not checking if the options are correct: Make sure to multiply the two binomials together and see if you get the original expression.
Not using the correct method: Make sure to use the correct method for factoring the expression, such as factoring by grouping or factoring by difference of squares.
Not paying attention to the coefficients: Make sure to pay attention to the coefficients of the expression and make sure they are correct.

Tips for Factoring Expressions

Here are a few tips for factoring expressions:

Use the correct method: Make sure to use the correct method for factoring the expression, such as factoring by grouping or factoring by difference of squares.
Pay attention to the coefficients: Make sure to pay attention to the coefficients of the expression and make sure they are correct.
Check your work: Make sure to check your work by multiplying the two binomials together and seeing if you get the original expression.

Conclusion

In this article, we discussed how to factor the expression $27y^2 + (-63 - 3a)y + 7a$ . We examined three different options and found that option C: $(9y-a)(3y-7)$ is the correct factorization. We also discussed common mistakes to avoid and tips for factoring expressions.