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

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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 explore the factored form of a given polynomial and provide a step-by-step guide on how to factor it.

What is Factoring a Polynomial?

Factoring a polynomial involves expressing it as a product of two or more polynomials. This is done by finding the factors of the polynomial, which are the numbers or expressions that multiply together to give the original polynomial. Factoring a polynomial can be useful in solving equations, graphing functions, and simplifying expressions.

The Given Polynomial

The given polynomial is:

x2−16x+48{ x^2 - 16x + 48 }

Factoring the Polynomial

To factor the polynomial, we need to find two numbers whose product is 48 and whose sum is -16. These numbers are -6 and -8, since (-6) × (-8) = 48 and (-6) + (-8) = -14, which is close to -16.

However, we can also try to factor the polynomial by grouping the terms. We can group the first two terms together and the last two terms together:

x2−16x+48=(x2−16x)+48{ x^2 - 16x + 48 = (x^2 - 16x) + 48 }

Factoring by Grouping

Now, we can factor out the common term from the first two terms:

(x2−16x)+48=x(x−16)+48{ (x^2 - 16x) + 48 = x(x - 16) + 48 }

Finding the Factors

Now, we need to find the factors of 48 that add up to -16. We can try different combinations of factors until we find the correct one.

After trying different combinations, we find that the factors of 48 that add up to -16 are -6 and -8.

Writing the Factored Form

Now that we have found the factors, we can write the factored form of the polynomial:

x2−16x+48=(x−6)(x−8){ x^2 - 16x + 48 = (x - 6)(x - 8) }

Conclusion

In this article, we have explored the factored form of a given polynomial and provided a step-by-step guide on how to factor it. We have also discussed the importance of factoring polynomials and how it can be useful in solving equations, graphing functions, and simplifying expressions.

Answer

The factored form of the polynomial is:

(x−6)(x−8){ (x - 6)(x - 8) }

This is option B in the given choices.

Tips and Tricks

Here are some tips and tricks to help you factor polynomials:

  • Look for common factors: Check if there are any common factors among the terms of the polynomial.
  • Use the distributive property: Use the distributive property to expand the polynomial and see if you can factor it.
  • Group the terms: Group the terms of the polynomial together and try to factor out common terms.
  • Use the factoring formulas: Use the factoring formulas, such as the difference of squares formula, to factor the polynomial.

Practice Problems

Here are some practice problems to help you practice factoring polynomials:

  • Factor the polynomial: x2+14x+40{ x^2 + 14x + 40 }
  • Factor the polynomial: x2−12x+36{ x^2 - 12x + 36 }
  • Factor the polynomial: x2+16x+64{ x^2 + 16x + 64 }

Conclusion

Factoring polynomials is an important concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we have explored the factored form of a given polynomial and provided a step-by-step guide on how to factor it. We have also discussed the importance of factoring polynomials and how it can be useful in solving equations, graphing functions, and simplifying expressions.

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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 on factoring polynomials, covering common questions and topics.

Q: What is factoring a polynomial?

A: Factoring a polynomial involves expressing it as a product of two or more polynomials. This is done by finding the factors of the polynomial, which are the numbers or expressions that multiply together to give the original polynomial.

Q: How do I factor a polynomial?

A: To factor a polynomial, you can try the following steps:

  1. Look for common factors: Check if there are any common factors among the terms of the polynomial.
  2. Use the distributive property: Use the distributive property to expand the polynomial and see if you can factor it.
  3. Group the terms: Group the terms of the polynomial together and try to factor out common terms.
  4. Use the factoring formulas: Use the factoring formulas, such as the difference of squares formula, to factor the polynomial.

Q: What are some common factoring formulas?

A: Some common factoring formulas include:

  • Difference of squares formula: a2−b2=(a+b)(a−b){ a^2 - b^2 = (a + b)(a - b) }
  • Sum of squares formula: a2+b2=(a+b)2−2ab{ a^2 + b^2 = (a + b)^2 - 2ab }
  • Difference of cubes formula: a3−b3=(a−b)(a2+ab+b2){ a^3 - b^3 = (a - b)(a^2 + ab + b^2) }

Q: How do I factor a quadratic polynomial?

A: To factor a quadratic polynomial, you can try the following steps:

  1. Look for two numbers whose product is the constant term: Find two numbers whose product is the constant term of the polynomial.
  2. Look for two numbers whose sum is the coefficient of the linear term: Find two numbers whose sum is the coefficient of the linear term of the polynomial.
  3. Write the factored form: Write the factored form of the polynomial using the two numbers you found.

Q: How do I factor a polynomial with a negative sign?

A: To factor a polynomial with a negative sign, you can try the following steps:

  1. Remove the negative sign: Remove the negative sign from the polynomial.
  2. Factor the polynomial: Factor the polynomial without the negative sign.
  3. Add a negative sign: Add a negative sign to 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 checking for common factors: Not checking for common factors among the terms of the polynomial.
  • Not using the distributive property: Not using the distributive property to expand the polynomial and see if you can factor it.
  • Not grouping the terms: Not grouping the terms of the polynomial together and trying to factor out common terms.

Q: How do I check if a polynomial is factored correctly?

A: To check if a polynomial is factored correctly, you can try the following steps:

  1. Multiply the factors: Multiply the factors of the polynomial together.
  2. Check if the product is the original polynomial: Check if the product is the original polynomial.
  3. Check if the factors are correct: Check if the factors are correct and if they multiply together to give the original polynomial.

Conclusion

Factoring polynomials is an important concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we have provided a Q&A guide on factoring polynomials, covering common questions and topics. We hope this guide has been helpful in understanding how to factor polynomials and avoiding common mistakes.