What Is The Factored Form Of The Polynomial X 2 − 16 X + 48 X^2 - 16x + 48 X 2 − 16 X + 48 ?A. ( X − 4 ) ( X − 12 (x - 4)(x - 12 ( X − 4 ) ( X − 12 ]B. ( X − 6 ) ( X − 8 (x - 6)(x - 8 ( X − 6 ) ( X − 8 ]C. ( X + 4 ) ( X + 12 (x + 4)(x + 12 ( X + 4 ) ( X + 12 ]D. ( X + 6 ) ( X + 8 (x + 6)(x + 8 ( X + 6 ) ( X + 8 ]

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Introduction

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we will focus on factoring the polynomial x2−16x+48x^2 - 16x + 48 and explore the different methods used to achieve this. We will also discuss the importance of factoring polynomials and provide examples of how it can be applied in real-world scenarios.

What is Factoring a Polynomial?

Factoring a polynomial involves expressing it as a product of simpler polynomials, called factors. These factors can be linear or quadratic expressions, and they can be combined in various ways to form the original polynomial. Factoring a polynomial can be useful in solving equations, graphing functions, and simplifying expressions.

Methods of Factoring Polynomials

There are several methods of factoring polynomials, including:

  • Factoring by Grouping: This method involves grouping the terms of the polynomial into pairs and factoring out the greatest common factor (GCF) from each pair.
  • Factoring by Difference of Squares: This method involves factoring a polynomial that can be expressed as the difference of two squares.
  • Factoring by Sum and Difference: This method involves factoring a polynomial that can be expressed as the sum or difference of two terms.

Factoring the Polynomial x2−16x+48x^2 - 16x + 48

To factor the polynomial x2−16x+48x^2 - 16x + 48, we can use the method of factoring by grouping. This involves grouping the terms of the polynomial into pairs and factoring out the GCF from each pair.

Step 1: Group the Terms

The polynomial x2−16x+48x^2 - 16x + 48 can be grouped into two pairs of terms:

  • x2−16xx^2 - 16x
  • 4848

Step 2: Factor Out the GCF

The GCF of the first pair of terms is xx, and the GCF of the second pair of terms is 4848. We can factor out the GCF from each pair of terms:

  • x(x−16)x(x - 16)
  • 4848

Step 3: Combine the Factors

We can combine the factors from each pair of terms to form the factored form of the polynomial:

  • (x−4)(x−12)(x - 4)(x - 12)

Conclusion

In this article, we have discussed the concept of factoring polynomials and explored the different methods used to achieve this. We have also factored the polynomial x2−16x+48x^2 - 16x + 48 using the method of factoring by grouping. The factored form of the polynomial is (x−4)(x−12)(x - 4)(x - 12).

Importance of Factoring Polynomials

Factoring polynomials is an important concept in algebra that has numerous applications in real-world scenarios. Some of the importance of factoring polynomials include:

  • Solving Equations: Factoring polynomials can be used to solve equations by setting each factor equal to zero and solving for the variable.
  • Graphing Functions: Factoring polynomials can be used to graph functions by identifying the x-intercepts and y-intercepts of the function.
  • Simplifying Expressions: Factoring polynomials can be used to simplify expressions by combining like terms and eliminating common factors.

Examples of Factoring Polynomials

Here are some examples of factoring polynomials:

  • Example 1: Factor the polynomial x2+5x+6x^2 + 5x + 6.
  • Solution: The factored form of the polynomial is (x+2)(x+3)(x + 2)(x + 3).
  • Example 2: Factor the polynomial x2−7x+12x^2 - 7x + 12.
  • Solution: The factored form of the polynomial is (x−3)(x−4)(x - 3)(x - 4).

Common Mistakes to Avoid

When factoring polynomials, there are several common mistakes to avoid, including:

  • Not Factoring Out the GCF: Failing to factor out the GCF from each pair of terms can lead to incorrect factored forms.
  • Not Combining Like Terms: Failing to combine like terms can lead to incorrect factored forms.
  • Not Checking the Factored Form: Failing to check the factored form can lead to incorrect solutions.

Conclusion

In conclusion, factoring polynomials is an important concept in algebra that has numerous applications in real-world scenarios. By understanding the different methods of factoring polynomials and avoiding common mistakes, we can simplify expressions, solve equations, and graph functions with ease.

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Introduction

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we will provide a Q&A guide to help you understand the concept of factoring polynomials and how to apply it in different scenarios.

Q: What is Factoring a Polynomial?

A: Factoring a polynomial involves expressing it as a product of simpler polynomials, called factors. These factors can be linear or quadratic expressions, and they can be combined in various ways to form the original polynomial.

Q: Why is Factoring a Polynomial Important?

A: Factoring a polynomial is important because it can be used to solve equations, graph functions, and simplify expressions. It is also a useful tool for identifying the x-intercepts and y-intercepts of a function.

Q: What are the Different Methods of Factoring Polynomials?

A: There are several methods of factoring polynomials, including:

  • Factoring by Grouping: This method involves grouping the terms of the polynomial into pairs and factoring out the greatest common factor (GCF) from each pair.
  • Factoring by Difference of Squares: This method involves factoring a polynomial that can be expressed as the difference of two squares.
  • Factoring by Sum and Difference: This method involves factoring a polynomial that can be expressed as the sum or difference of two terms.

Q: How Do I Factor a Polynomial Using the Method of Factoring by Grouping?

A: To factor a polynomial using the method of factoring by grouping, follow these steps:

  1. Group the terms of the polynomial into pairs.
  2. Factor out the GCF from each pair of terms.
  3. Combine the factors from each pair of terms to form the factored form of the polynomial.

Q: How Do I Factor a Polynomial Using the Method of Factoring by Difference of Squares?

A: To factor a polynomial using the method of factoring by difference of squares, follow these steps:

  1. Identify the polynomial as a difference of two squares.
  2. Factor the polynomial as the product of two binomials.
  3. Simplify the expression to form the factored form of the polynomial.

Q: How Do I Factor a Polynomial Using the Method of Factoring by Sum and Difference?

A: To factor a polynomial using the method of factoring by sum and difference, follow these steps:

  1. Identify the polynomial as a sum or difference of two terms.
  2. Factor the polynomial as the product of two binomials.
  3. Simplify the expression to form the factored form of the polynomial.

Q: What are Some Common Mistakes to Avoid When Factoring Polynomials?

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

  • Not Factoring Out the GCF: Failing to factor out the GCF from each pair of terms can lead to incorrect factored forms.
  • Not Combining Like Terms: Failing to combine like terms can lead to incorrect factored forms.
  • Not Checking the Factored Form: Failing to check the factored form can lead to incorrect solutions.

Q: How Do I Check the Factored Form of a Polynomial?

A: To check the factored form of a polynomial, follow these steps:

  1. Multiply the factors together to form the original polynomial.
  2. Simplify the expression to ensure that it is equal to the original polynomial.
  3. Verify that the factored form is correct.

Conclusion

In conclusion, factoring polynomials is an important concept in algebra that has numerous applications in real-world scenarios. By understanding the different methods of factoring polynomials and avoiding common mistakes, we can simplify expressions, solve equations, and graph functions with ease.