Choose The Correct Complete Factorization Of $z^4-16$.A. $\left(z^2+4\right)(z-2)(z+2$\]B. $(z-2)^4$C. $(z-2)^2(z+2)^2$D. $\left(z^2-4\right)^2$E. $\left(z^2+2\right)\left(z^2-2\right$\]

Feb 25, 2025 by ADMIN 189 views

Introduction

In this article, we will explore the concept of complete factorization and apply it to the given expression $z^4-16$ . The complete factorization of an algebraic expression is a way of expressing it as a product of simpler expressions, called factors. This is a fundamental concept in algebra and is used extensively in various mathematical disciplines.

Understanding the Expression

The given expression is $z^4-16$ . This is a difference of squares, which can be factored using the formula $a^2-b^2=(a+b)(a-b)$ . In this case, we have $a=z^2$ and $b=4$ . Therefore, we can write $z^4-16$ as $(z^2+4)(z^2-4)$ .

Factoring the Expression

Now, we need to factor the expression $(z^2-4)$ . This is also a difference of squares, and we can factor it as $(z-2)(z+2)$ . Therefore, the complete factorization of $z^4-16$ is $(z^2+4)(z-2)(z+2)$ .

Comparing with the Options

Let's compare our result with the options given:

A. $\left(z^2+4\right)(z-2)(z+2)$ B. $(z-2)^4$ C. $(z-2)^2(z+2)^2$ D. $\left(z^2-4\right)^2$ E. $\left(z^2+2\right)\left(z^2-2\right)$

Our result matches option A.

Why is Option A Correct?

Option A is correct because it correctly factors the expression $z^4-16$ as a product of simpler expressions. The factor $(z^2+4)$ is a difference of squares, and the factors $(z-2)$ and $(z+2)$ are also correct.

Why are the Other Options Incorrect?

Option B is incorrect because it factors the expression as $(z-2)^4$ , which is not correct. The correct factorization is $(z-2)(z+2)$ , not $(z-2)^4$ .

Option C is incorrect because it factors the expression as $(z-2)^2(z+2)^2$ , which is not correct. The correct factorization is $(z-2)(z+2)$ , not $(z-2)^2(z+2)^2$ .

Option D is incorrect because it factors the expression as $\left(z^2-4\right)^2$ , which is not correct. The correct factorization is $(z^2-4)$ , not $\left(z^2-4\right)^2$ .

Option E is incorrect because it factors the expression as $\left(z^2+2\right)\left(z^2-2\right)$ , which is not correct. The correct factorization is $(z^2+4)(z-2)(z+2)$ , not $\left(z^2+2\right)\left(z^2-2\right)$ .

Conclusion

Final Answer

The final answer is $\boxed{\left(z^2+4\right)(z-2)(z+2)}$ .

Additional Tips and Tricks

When factoring an expression, always look for the difference of squares formula.
Use the formula $a^2-b^2=(a+b)(a-b)$ to factor the expression.
Make sure to check your work by multiplying the factors together to get the original expression.

Common Mistakes to Avoid

Don't forget to use the difference of squares formula when factoring an expression.
Make sure to check your work by multiplying the factors together to get the original expression.
Don't factor an expression as a product of simpler expressions if it's not correct.

Real-World Applications

The concept of complete factorization is used extensively in various mathematical disciplines, such as algebra, geometry, and calculus.
It's used to solve equations and inequalities, and to find the roots of a polynomial.
It's also used in computer science and engineering to optimize algorithms and solve problems.

Final Thoughts

In conclusion, the correct complete factorization of $z^4-16$ is $\left(z^2+4\right)(z-2)(z+2)$ . This is because it correctly factors the expression as a product of simpler expressions, using the difference of squares formula. We hope this article has helped you understand the concept of complete factorization and how to apply it to solve problems.

Q: What is complete factorization?

A: Complete factorization is a way of expressing an algebraic expression as a product of simpler expressions, called factors. This is a fundamental concept in algebra and is used extensively in various mathematical disciplines.

Q: How do I factor an expression using the difference of squares formula?

A: To factor an expression using the difference of squares formula, you need to identify the two perfect squares that make up the expression. The formula is $a^2-b^2=(a+b)(a-b)$ . For example, if you have the expression $z^4-16$ , you can factor it as $(z^2+4)(z^2-4)$ .

Q: What is the difference of squares formula?

A: The difference of squares formula is $a^2-b^2=(a+b)(a-b)$ . This formula can be used to factor expressions that are in the form of a difference of squares.

Q: How do I check my work when factoring an expression?

A: To check your work when factoring an expression, you need to multiply the factors together to get the original expression. For example, if you have the expression $(z^2+4)(z-2)(z+2)$ , you can multiply the factors together to get $z^4-16$ .

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

A: Some common mistakes to avoid when factoring an expression include:

Not using the difference of squares formula when factoring an expression
Not checking your work by multiplying the factors together to get the original expression
Factoring an expression as a product of simpler expressions if it's not correct

Q: How is complete factorization used in real-world applications?

A: Complete factorization is used extensively in various mathematical disciplines, such as algebra, geometry, and calculus. It's used to solve equations and inequalities, and to find the roots of a polynomial. It's also used in computer science and engineering to optimize algorithms and solve problems.

Q: What are some tips and tricks for factoring expressions?

A: Some tips and tricks for factoring expressions include:

Always look for the difference of squares formula when factoring an expression
Use the formula $a^2-b^2=(a+b)(a-b)$ to factor the expression
Make sure to check your work by multiplying the factors together to get the original expression

Q: How do I know if an expression can be factored using the difference of squares formula?

A: To determine if an expression can be factored using the difference of squares formula, you need to check if the expression is in the form of a difference of squares. If it is, then you can use the formula to factor the expression.

Q: What are some examples of expressions that can be factored using the difference of squares formula?

A: Some examples of expressions that can be factored using the difference of squares formula include:

$z^4-16$
$x^2-9$
$y^2-25$

Q: How do I factor expressions that are not in the form of a difference of squares?

A: To factor expressions that are not in the form of a difference of squares, you need to use other factoring techniques, such as factoring by grouping or factoring by synthetic division.

Q: What are some common expressions that can be factored using the difference of squares formula?

A: Some common expressions that can be factored using the difference of squares formula include:

$z^4-16$
$x^2-9$
$y^2-25$
$a^2-b^2$
$c^2-d^2$

Q: How do I know if an expression can be factored using the difference of squares formula?

Q: What are some examples of expressions that cannot be factored using the difference of squares formula?

A: Some examples of expressions that cannot be factored using the difference of squares formula include:

$z^3-27$
$x^2+9$
$y^2+25$

Q: How do I factor expressions that cannot be factored using the difference of squares formula?

A: To factor expressions that cannot be factored using the difference of squares formula, you need to use other factoring techniques, such as factoring by grouping or factoring by synthetic division.

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

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

Not using the difference of squares formula when factoring an expression
Not checking your work by multiplying the factors together to get the original expression
Factoring an expression as a product of simpler expressions if it's not correct

Q: How do I know if an expression can be factored using the difference of squares formula?

Q: What are some examples of expressions that can be factored using the difference of squares formula?

A: Some examples of expressions that can be factored using the difference of squares formula include:

$z^4-16$
$x^2-9$
$y^2-25$

Q: How do I factor expressions that are not in the form of a difference of squares?

A: To factor expressions that are not in the form of a difference of squares, you need to use other factoring techniques, such as factoring by grouping or factoring by synthetic division.

Q: What are some common expressions that can be factored using the difference of squares formula?

A: Some common expressions that can be factored using the difference of squares formula include:

$z^4-16$
$x^2-9$
$y^2-25$
$a^2-b^2$
$c^2-d^2$

Q: How do I know if an expression can be factored using the difference of squares formula?

Q: What are some examples of expressions that cannot be factored using the difference of squares formula?

A: Some examples of expressions that cannot be factored using the difference of squares formula include:

$z^3-27$
$x^2+9$
$y^2+25$

Q: How do I factor expressions that cannot be factored using the difference of squares formula?

A: To factor expressions that cannot be factored using the difference of squares formula, you need to use other factoring techniques, such as factoring by grouping or factoring by synthetic division.

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

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

Not using the difference of squares formula when factoring an expression
Not checking your work by multiplying the factors together to get the original expression
Factoring an expression as a product of simpler expressions if it's not correct