Factor The Expression: Z 3 + 17 Z 2 + 70 Z Z^3 + 17z^2 + 70z Z 3 + 17 Z 2 + 70 Z Show Your Work Here:

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Introduction

Factoring an algebraic expression is a crucial skill in mathematics, and it is essential to understand the different techniques used to factor various types of expressions. In this article, we will focus on factoring the expression $z^3 + 17z^2 + 70z$. We will show the step-by-step process of factoring this expression and provide a clear explanation of the techniques used.

Understanding the Expression

Before we begin factoring the expression, let's take a closer look at it. The expression is a cubic polynomial, which means it has three terms. The first term is $z^3$, the second term is $17z^2$, and the third term is $70z$. We can see that the expression has a common factor of $z$, which is the highest power of $z$ that divides all three terms.

Factoring Out the Common Factor

The first step in factoring the expression is to factor out the common factor of $z$. We can do this by dividing each term by $z$. This will give us:

$$z^3 + 17z^2 + 70z = z(z^2 + 17z + 70)$$

We have factored out the common factor of $z$, and now we are left with a quadratic expression inside the parentheses.

Factoring the Quadratic Expression

The quadratic expression inside the parentheses is $z^2 + 17z + 70$. We can factor this expression by finding two numbers whose product is $70$ and whose sum is $17$. These numbers are $10$ and $7$, because $10 \times 7 = 70$ and $10 + 7 = 17$.

We can write the quadratic expression as:

$$z^2 + 17z + 70 = (z + 10)(z + 7)$$

Now we have factored the quadratic expression into two binomial factors.

Factoring the Entire Expression

We can now factor the entire expression by multiplying the common factor of $z$ with the factored quadratic expression:

$$z^3 + 17z^2 + 70z = z(z + 10)(z + 7)$$

We have successfully factored the expression $z^3 + 17z^2 + 70z$.

Conclusion

Factoring an algebraic expression is a crucial skill in mathematics, and it is essential to understand the different techniques used to factor various types of expressions. In this article, we have shown the step-by-step process of factoring the expression $z^3 + 17z^2 + 70z$. We have factored out the common factor of $z$ and then factored the quadratic expression inside the parentheses. We have also provided a clear explanation of the techniques used and have shown the final factored form of the expression.

Common Factors and Factoring

When factoring an algebraic expression, it is essential to look for common factors. A common factor is a factor that divides all the terms of the expression. In the case of the expression $z^3 + 17z^2 + 70z$, the common factor is $z$. We can factor out the common factor by dividing each term by $z$.

Quadratic Expressions and Factoring

Quadratic expressions are expressions of the form $ax^2 + bx + c$. We can factor quadratic expressions by finding two numbers whose product is $ac$ and whose sum is $b$. These numbers are called the roots of the quadratic expression. In the case of the expression $z^2 + 17z + 70$, the roots are $10$ and $7$.

Binomial Factors and Factoring

Binomial factors are factors of the form $(x + a)$. We can factor expressions by multiplying binomial factors. In the case of the expression $z^3 + 17z^2 + 70z$, we can factor it by multiplying the common factor of $z$ with the factored quadratic expression.

Real-World Applications of Factoring

Factoring is a crucial skill in mathematics, and it has many real-world applications. In algebra, factoring is used to solve equations and inequalities. In calculus, factoring is used to find the derivative of a function. In physics, factoring is used to solve problems involving motion and energy.

Conclusion

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Q: What is the first step in factoring the expression $z^3 + 17z^2 + 70z$?

A: The first step in factoring the expression $z^3 + 17z^2 + 70z$ is to factor out the common factor of $z$. We can do this by dividing each term by $z$.

Q: Why is it important to factor out the common factor of $z$?

A: Factoring out the common factor of $z$ is important because it allows us to simplify the expression and make it easier to factor. By factoring out the common factor, we can reduce the expression to a quadratic expression, which is easier to factor.

Q: How do we factor the quadratic expression $z^2 + 17z + 70$?

A: We can factor the quadratic expression $z^2 + 17z + 70$ by finding two numbers whose product is $70$ and whose sum is $17$. These numbers are $10$ and $7$, because $10 \times 7 = 70$ and $10 + 7 = 17$. We can write the quadratic expression as $(z + 10)(z + 7)$.

Q: What is the final factored form of the expression $z^3 + 17z^2 + 70z$?

A: The final factored form of the expression $z^3 + 17z^2 + 70z$ is $z(z + 10)(z + 7)$.

Q: Why is factoring important in mathematics?

A: Factoring is important in mathematics because it allows us to simplify complex expressions and make them easier to work with. Factoring is also used to solve equations and inequalities, and it has many real-world applications in fields such as physics and engineering.

Q: What are some common mistakes to avoid when factoring expressions?

A: Some common mistakes to avoid when factoring expressions include:

• Not factoring out the common factor of $z$
• Not finding the correct roots of the quadratic expression
• Not multiplying the binomial factors correctly
• Not checking the final factored form for errors

Q: How can I practice factoring expressions?

A: You can practice factoring expressions by working through examples and exercises in your textbook or online resources. You can also try factoring expressions on your own, using different techniques and strategies to see what works best for you.

Q: What are some real-world applications of factoring?

A: Some real-world applications of factoring include:

• Solving equations and inequalities in physics and engineering
• Finding the derivative of a function in calculus
• Simplifying complex expressions in algebra
• Solving problems involving motion and energy in physics

Q: Can you provide more examples of factoring expressions?

A: Yes, here are a few more examples of factoring expressions:

• $x^2 + 5x + 6 = (x + 3)(x + 2)$
• $y^2 - 4y - 5 = (y - 5)(y + 1)$
• $z^2 + 2z - 15 = (z + 5)(z - 3)$

I hope these examples help you to understand the concept of factoring expressions better. If you have any more questions or need further clarification, please don't hesitate to ask.