What Is The Completely Factored Form Of $x^2 - 16xy + 64y^2$?A. X Y ( X − 16 + 64 Y Xy(x - 16 + 64y X Y ( X − 16 + 64 Y ] B. X Y ( X + 16 + 64 Y Xy(x + 16 + 64y X Y ( X + 16 + 64 Y ] C. ( X − 8 Y ) ( X − 8 Y (x - 8y)(x - 8y ( X − 8 Y ) ( X − 8 Y ] D. ( X + 8 Y ) ( X + 8 Y (x + 8y)(x + 8y ( X + 8 Y ) ( X + 8 Y ]

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Introduction

In mathematics, factoring is a fundamental concept that involves expressing an algebraic expression as a product of simpler expressions. The completely factored form of an expression is the most simplified form, where the expression is written as a product of prime factors. In this article, we will explore the completely factored form of the given quadratic expression $x^2 - 16xy + 64y^2$.

Understanding the Expression

The given expression is a quadratic expression in the form of $ax^2 + bx + c$. In this case, the coefficients are $a = 1$, $b = -16y$, and $c = 64y^2$. To factorize this expression, we need to find two binomials whose product is equal to the given expression.

Factoring the Expression

To factorize the expression, we can use the method of splitting the middle term. This involves finding two numbers whose product is equal to the product of the coefficients of the first and last terms, and whose sum is equal to the coefficient of the middle term.

In this case, the product of the coefficients of the first and last terms is $1 \times 64y^2 = 64y^2$. The coefficient of the middle term is $-16y$. We need to find two numbers whose product is $64y^2$ and whose sum is $-16y$.

Finding the Numbers

Let's find the two numbers whose product is $64y^2$ and whose sum is $-16y$. We can start by listing the factors of $64y^2$:

• $1 \times 64y^2$
• $2 \times 32y^2$
• $4 \times 16y^2$
• $8 \times 8y^2$

We can see that the last pair of factors, $8 \times 8y^2$, has a sum of $16y^2$. However, this is not equal to $-16y$. We need to find another pair of factors whose sum is $-16y$.

Finding the Correct Pair

After re-examining the factors, we can see that the pair $-8y \times -8y$ has a product of $64y^2$ and a sum of $-16y$. This is the correct pair of numbers that we need to use to factorize the expression.

Factoring the Expression

Now that we have found the correct pair of numbers, we can factorize the expression as follows:

$x^2 - 16xy + 64y^2 = (x - 8y)(x - 8y)$

Conclusion

In this article, we have explored the completely factored form of the given quadratic expression $x^2 - 16xy + 64y^2$. We have used the method of splitting the middle term to factorize the expression, and have found that the completely factored form is $(x - 8y)(x - 8y)$.

Answer

The completely factored form of $x^2 - 16xy + 64y^2$ is:

$(x - 8y)(x - 8y)$

This matches option C in the given multiple-choice question.

Discussion

The completely factored form of a quadratic expression is an important concept in mathematics. It allows us to simplify complex expressions and solve equations more easily. In this article, we have seen how to factorize a quadratic expression using the method of splitting the middle term.

Tips and Tricks

• When factoring a quadratic expression, always look for two numbers whose product is equal to the product of the coefficients of the first and last terms, and whose sum is equal to the coefficient of the middle term.
• Use the method of splitting the middle term to factorize the expression.
• Check your work by multiplying the two binomials together to ensure that you get the original expression.

Related Topics

• Factoring quadratic expressions
• Solving quadratic equations
• Algebraic expressions

References

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

Introduction

In our previous article, we explored the completely factored form of the quadratic expression $x^2 - 16xy + 64y^2$. We also discussed the method of splitting the middle term to factorize the expression. In this article, we will answer some frequently asked questions (FAQs) about factoring quadratic expressions.

Q: What is the difference between factoring and simplifying an expression?

A: Factoring an expression involves expressing it as a product of simpler expressions, while simplifying an expression involves combining like terms to reduce its complexity.

Q: How do I know if an expression can be factored?

A: If an expression can be factored, it will have a specific pattern or structure that allows it to be expressed as a product of simpler expressions. For example, if an expression has a common factor, it can be factored out.

Q: What is the difference between factoring a quadratic expression and factoring a polynomial expression?

A: Factoring a quadratic expression involves expressing it as a product of two binomials, while factoring a polynomial expression involves expressing it as a product of simpler expressions, which may be binomials, trinomials, or other types of expressions.

Q: How do I factor a quadratic expression with a negative coefficient?

A: To factor a quadratic expression with a negative coefficient, you can use the method of splitting the middle term, just like you would for a quadratic expression with a positive coefficient.

Q: Can I factor a quadratic expression with a variable in the coefficient?

A: Yes, you can factor a quadratic expression with a variable in the coefficient. However, you may need to use a different method, such as the method of substitution or the method of elimination.

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

A: To check your work, multiply the two binomials together to ensure that you get the original expression. If the product is not equal to the original expression, you may need to re-factor the expression.

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

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

• Not checking your work
• Not using the correct method for factoring
• Not factoring out common factors
• Not simplifying the expression after factoring

Q: Can I use technology to help me factor quadratic expressions?

A: Yes, you can use technology, such as calculators or computer software, to help you factor quadratic expressions. However, it's still important to understand the underlying math and to be able to factor expressions by hand.

Q: How do I factor a quadratic expression with a complex coefficient?

A: To factor a quadratic expression with a complex coefficient, you can use the method of splitting the middle term, just like you would for a quadratic expression with a real coefficient.

Q: Can I factor a quadratic expression with a variable in the exponent?

A: No, you cannot factor a quadratic expression with a variable in the exponent. However, you may be able to simplify the expression using other methods, such as the method of substitution or the method of elimination.

Conclusion

In this article, we have answered some frequently asked questions (FAQs) about factoring quadratic expressions. We have discussed the method of splitting the middle term, common mistakes to avoid, and how to check your work. We have also explored some advanced topics, such as factoring quadratic expressions with complex coefficients and variables in the exponent.

Tips and Tricks

• Always check your work when factoring a quadratic expression.
• Use the method of splitting the middle term to factor quadratic expressions.
• Factor out common factors to simplify the expression.
• Use technology, such as calculators or computer software, to help you factor quadratic expressions.

Related Topics

• Factoring polynomial expressions
• Simplifying algebraic expressions
• Solving quadratic equations

References

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