Rewrite The Expression By Factoring Out { (y-3)$} . . . { 8y^2(y-3) + (y-3) \}

Introduction

In algebra, factoring is a powerful technique used to simplify complex expressions and solve equations. One common method of factoring involves factoring out a common binomial, which is a polynomial expression with two terms. In this article, we will explore how to factor out a common binomial using the expression $8y^2(y-3) + (y-3)$ as an example.

Understanding Common Binomials

A common binomial is a polynomial expression that can be factored out of a larger expression. It is a binomial, meaning it has two terms, and it is common, meaning it appears in every term of the larger expression. In the expression $8y^2(y-3) + (y-3)$, the common binomial is $(y-3)$.

Factoring Out the Common Binomial

To factor out the common binomial $(y-3)$, we need to identify the terms that contain this binomial. In the expression $8y^2(y-3) + (y-3)$, both terms contain the binomial $(y-3)$. We can factor out this binomial by grouping the terms and using the distributive property.

Step 1: Identify the Terms with the Common Binomial

The first step in factoring out the common binomial is to identify the terms that contain this binomial. In the expression $8y^2(y-3) + (y-3)$, both terms contain the binomial $(y-3)$.

Step 2: Group the Terms

Next, we group the terms that contain the common binomial. In this case, we can group the two terms as follows:

$$8y^2(y-3) + (y-3) = (8y^2 + 1)(y-3)$$

Step 3: Factor Out the Common Binomial

Now that we have grouped the terms, we can factor out the common binomial. To do this, we use the distributive property, which states that for any numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. In this case, we can factor out the common binomial $(y-3)$ as follows:

$$(8y^2 + 1)(y-3) = (y-3)(8y^2 + 1)$$

Step 4: Simplify the Expression

Finally, we can simplify the expression by combining the terms. In this case, we can combine the terms as follows:

$$(y-3)(8y^2 + 1) = 8y^3 - 24y^2 + y - 3$$

Conclusion

Factoring out a common binomial is a powerful technique used to simplify complex expressions and solve equations. By identifying the terms that contain the common binomial, grouping the terms, factoring out the common binomial, and simplifying the expression, we can simplify complex expressions and solve equations. In this article, we used the expression $8y^2(y-3) + (y-3)$ as an example to demonstrate how to factor out a common binomial.

Common Binomial Factoring Examples

Here are some examples of factoring out common binomials:

• $3x^2(x+2) + (x+2) = (3x^2 + 1)(x+2)$
• $4y^3(y-2) + (y-2) = (4y^3 + 1)(y-2)$
• $2x^2(x-1) + (x-1) = (2x^2 + 1)(x-1)$

Tips and Tricks

Here are some tips and tricks for factoring out common binomials:

• Identify the terms that contain the common binomial.
• Group the terms that contain the common binomial.
• Factor out the common binomial using the distributive property.
• Simplify the expression by combining the terms.

Conclusion

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Introduction

In our previous article, we explored how to factor out a common binomial using the expression $8y^2(y-3) + (y-3)$ as an example. In this article, we will answer some frequently asked questions about factoring out common binomials.

Q&A

Q: What is a common binomial?

A: A common binomial is a polynomial expression with two terms that can be factored out of a larger expression.

Q: How do I identify the terms that contain the common binomial?

A: To identify the terms that contain the common binomial, look for terms that have the same binomial factor. In the expression $8y^2(y-3) + (y-3)$, both terms contain the binomial $(y-3)$.

Q: How do I group the terms that contain the common binomial?

A: To group the terms that contain the common binomial, combine the terms that have the same binomial factor. In the expression $8y^2(y-3) + (y-3)$, we can group the two terms as follows:

$$8y^2(y-3) + (y-3) = (8y^2 + 1)(y-3)$$

Q: How do I factor out the common binomial?

A: To factor out the common binomial, use the distributive property, which states that for any numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. In this case, we can factor out the common binomial $(y-3)$ as follows:

$$(8y^2 + 1)(y-3) = (y-3)(8y^2 + 1)$$

Q: What is the difference between factoring out a common binomial and factoring a quadratic expression?

A: Factoring out a common binomial involves factoring out a binomial factor from a larger expression, whereas factoring a quadratic expression involves factoring a quadratic expression into the product of two binomials.

Q: Can I factor out a common binomial from any expression?

A: No, you cannot factor out a common binomial from any expression. The expression must have a common binomial factor for you to factor it out.

Q: How do I simplify the expression after factoring out the common binomial?

A: To simplify the expression after factoring out the common binomial, combine the terms. In the expression $(y-3)(8y^2 + 1)$, we can simplify the expression as follows:

$$(y-3)(8y^2 + 1) = 8y^3 - 24y^2 + y - 3$$

Common Binomial Factoring Examples

Here are some examples of factoring out common binomials:

• $3x^2(x+2) + (x+2) = (3x^2 + 1)(x+2)$
• $4y^3(y-2) + (y-2) = (4y^3 + 1)(y-2)$
• $2x^2(x-1) + (x-1) = (2x^2 + 1)(x-1)$

Tips and Tricks

Here are some tips and tricks for factoring out common binomials:

• Identify the terms that contain the common binomial.
• Group the terms that contain the common binomial.
• Factor out the common binomial using the distributive property.
• Simplify the expression by combining the terms.

Conclusion

Factoring out a common binomial is a powerful technique used to simplify complex expressions and solve equations. By identifying the terms that contain the common binomial, grouping the terms, factoring out the common binomial, and simplifying the expression, we can simplify complex expressions and solve equations. In this article, we answered some frequently asked questions about factoring out common binomials.