Simplify The Expression: $y^3 + 7y^2 - Y - 7$

$$

Introduction

In this article, we will simplify the given expression $y^3 + 7y^2 - y - 7$. This involves factoring the expression to its simplest form, which will make it easier to work with and understand. We will use various algebraic techniques to simplify the expression.

Step 1: Factor out the Greatest Common Factor (GCF)

The first step in simplifying the expression is to factor out the greatest common factor (GCF). The GCF is the largest factor that divides all the terms in the expression. In this case, the GCF is 1, since there is no common factor that divides all the terms.

However, we can try to factor out a common factor from two or more terms. Let's examine the expression closely. We can see that the terms $y^3$ and $7y^2$ have a common factor of $y^2$, and the terms $-y$ and $-7$ have a common factor of $-1$. We can factor out these common factors as follows:

$$y^3 + 7y^2 - y - 7 = y^2(y + 7) - 1(y + 7)$$

Step 2: Factor the Expression using the Difference of Squares Formula

Now that we have factored out the common factors, we can use the difference of squares formula to factor the expression further. The difference of squares formula states that:

$$a^2 - b^2 = (a + b)(a - b)$$

We can see that the expression $y^2(y + 7) - 1(y + 7)$ can be written as:

$$(y^2 - 1)(y + 7)$$

Step 3: Simplify the Expression using the Difference of Squares Formula

Now that we have factored the expression using the difference of squares formula, we can simplify it further. We can see that the expression $(y^2 - 1)(y + 7)$ can be written as:

$$(y - 1)(y + 1)(y + 7)$$

Conclusion

In this article, we simplified the expression $y^3 + 7y^2 - y - 7$ using various algebraic techniques. We factored out the greatest common factor (GCF), and then used the difference of squares formula to factor the expression further. Finally, we simplified the expression to its simplest form, which is $(y - 1)(y + 1)(y + 7)$.

Example Use Cases

The simplified expression $(y - 1)(y + 1)(y + 7)$ can be used in various mathematical applications, such as:

• Finding the roots of the equation $y^3 + 7y^2 - y - 7 = 0$
• Simplifying complex algebraic expressions
• Solving systems of equations

Tips and Tricks

When simplifying expressions, it's essential to look for common factors and use algebraic formulas to factor the expression. In this case, we used the difference of squares formula to factor the expression further. Additionally, we can use other algebraic formulas, such as the sum of cubes formula, to simplify expressions.

Common Mistakes to Avoid

When simplifying expressions, it's essential to avoid common mistakes, such as:

• Not factoring out the greatest common factor (GCF)
• Not using algebraic formulas to factor the expression
• Not simplifying the expression to its simplest form

By following these tips and tricks, and avoiding common mistakes, you can simplify expressions like $y^3 + 7y^2 - y - 7$ with ease.

Glossary of Terms

• Greatest Common Factor (GCF): The largest factor that divides all the terms in an expression.
• Difference of Squares Formula: A formula that states that $a^2 - b^2 = (a + b)(a - b)$.
• Sum of Cubes Formula: A formula that states that $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton
  Simplify the Expression: $y^3 + 7y^2 - y - 7$ - Q&A =====================================================

Introduction

In our previous article, we simplified the expression $y^3 + 7y^2 - y - 7$ using various algebraic techniques. In this article, we will answer some frequently asked questions (FAQs) related to simplifying expressions like this one.

Q: What is the greatest common factor (GCF) of the expression $y^3 + 7y^2 - y - 7$?

A: The greatest common factor (GCF) of the expression $y^3 + 7y^2 - y - 7$ is 1, since there is no common factor that divides all the terms.

Q: How do I factor out the greatest common factor (GCF) from the expression $y^3 + 7y^2 - y - 7$?

A: To factor out the greatest common factor (GCF) from the expression $y^3 + 7y^2 - y - 7$, you can look for common factors among two or more terms. In this case, we can factor out a common factor of $y^2$ from the terms $y^3$ and $7y^2$, and a common factor of $-1$ from the terms $-y$ and $-7$. This gives us:

$$y^3 + 7y^2 - y - 7 = y^2(y + 7) - 1(y + 7)$$

Q: How do I use the difference of squares formula to factor the expression $y^2(y + 7) - 1(y + 7)$?

A: To use the difference of squares formula to factor the expression $y^2(y + 7) - 1(y + 7)$, you can rewrite it as:

$$(y^2 - 1)(y + 7)$$

Then, you can use the difference of squares formula to factor the expression further:

$$(y - 1)(y + 1)(y + 7)$$

Q: What are some common mistakes to avoid when simplifying expressions like $y^3 + 7y^2 - y - 7$?

A: Some common mistakes to avoid when simplifying expressions like $y^3 + 7y^2 - y - 7$ include:

• Not factoring out the greatest common factor (GCF)
• Not using algebraic formulas to factor the expression
• Not simplifying the expression to its simplest form

Q: How do I use the simplified expression $(y - 1)(y + 1)(y + 7)$ in real-world applications?

A: The simplified expression $(y - 1)(y + 1)(y + 7)$ can be used in various real-world applications, such as:

• Finding the roots of the equation $y^3 + 7y^2 - y - 7 = 0$
• Simplifying complex algebraic expressions
• Solving systems of equations

Q: What are some other algebraic formulas that I can use to simplify expressions like $y^3 + 7y^2 - y - 7$?

A: Some other algebraic formulas that you can use to simplify expressions like $y^3 + 7y^2 - y - 7$ include:

• The sum of cubes formula: $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$
• The difference of cubes formula: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$

Conclusion

In this article, we answered some frequently asked questions (FAQs) related to simplifying expressions like $y^3 + 7y^2 - y - 7$. We covered topics such as factoring out the greatest common factor (GCF), using the difference of squares formula, and avoiding common mistakes. We also discussed some real-world applications of the simplified expression $(y - 1)(y + 1)(y + 7)$ and some other algebraic formulas that you can use to simplify expressions like this one.

Glossary of Terms

• Greatest Common Factor (GCF): The largest factor that divides all the terms in an expression.
• Difference of Squares Formula: A formula that states that $a^2 - b^2 = (a + b)(a - b)$.
• Sum of Cubes Formula: A formula that states that $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$.
• Difference of Cubes Formula: A formula that states that $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$.

References

• [1] "Algebra" by Michael Artin
• [2] "Calculus" by Michael Spivak
• [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton