Factor The Polynomial Completely. 9 V 2 + 9 V − 648 9v^2 + 9v - 648 9 V 2 + 9 V − 648 A. 9 ( V + 8 ) ( V − 9 9(v+8)(v-9 9 ( V + 8 ) ( V − 9 ]B. ( 9 V − 8 ) ( 9 V + 9 (9v-8)(9v+9 ( 9 V − 8 ) ( 9 V + 9 ]C. ( 9 V + 8 ) ( V − 9 (9v+8)(v-9 ( 9 V + 8 ) ( V − 9 ]D. 9 ( V − 8 ) ( V + 9 9(v-8)(v+9 9 ( V − 8 ) ( V + 9 ]

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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 9v2+9v−6489v^2 + 9v - 648 completely. We will explore the different methods and techniques used to factor polynomials, and provide a step-by-step guide on how to factor the given polynomial.

Understanding the Polynomial

Before we begin factoring the polynomial, it's essential to understand its structure and properties. The given polynomial is a quadratic polynomial, which means it has a degree of 2. It can be written in the general form ax2+bx+cax^2 + bx + c, where aa, bb, and cc are constants.

In this case, the polynomial is 9v2+9v−6489v^2 + 9v - 648. We can see that the coefficient of the v2v^2 term is 9, the coefficient of the vv term is 9, and the constant term is -648.

Factoring the Polynomial

To factor the polynomial completely, we need to find two binomials whose product is equal to the given polynomial. We can start by looking for two numbers whose product is equal to the product of the coefficient of the v2v^2 term and the constant term, and whose sum is equal to the coefficient of the vv term.

In this case, we need to find two numbers whose product is equal to 9×−648=−58329 \times -648 = -5832, and whose sum is equal to 9. After some trial and error, we find that the numbers are 72 and -81.

Using the Factoring Method

Now that we have found the two numbers, we can use the factoring method to factor the polynomial. We can write the polynomial as a product of two binomials, where the first binomial has the coefficient of the v2v^2 term and the second binomial has the two numbers we found.

Using the factoring method, we can write the polynomial as:

9v2+9v−648=(9v+72)(9v−81)9v^2 + 9v - 648 = (9v + 72)(9v - 81)

Checking the Factored Form

To check if the factored form is correct, we can multiply the two binomials together and see if we get the original polynomial.

(9v+72)(9v−81)=9v2−729v+648v−5832(9v + 72)(9v - 81) = 9v^2 - 729v + 648v - 5832

Simplifying the expression, we get:

9v2−81v−58329v^2 - 81v - 5832

This is not equal to the original polynomial, so we need to try again.

Alternative Factoring Method

Another method to factor the polynomial is to use the grouping method. We can group the terms of the polynomial into two pairs, and then factor each pair separately.

Using the grouping method, we can write the polynomial as:

9v2+9v−648=(9v2+9v)−(648)9v^2 + 9v - 648 = (9v^2 + 9v) - (648)

Factoring each pair separately, we get:

(9v2+9v)=9v(v+1)(9v^2 + 9v) = 9v(v + 1)

−(648)=−648-(648) = -648

Now, we can combine the two expressions to get:

9v2+9v−648=9v(v+1)−6489v^2 + 9v - 648 = 9v(v + 1) - 648

Factoring the Difference of Squares

We can see that the expression 9v2+9v9v^2 + 9v is a difference of squares, which can be factored as:

9v2+9v=(3v)2+3v=(3v+9)(3v−9)9v^2 + 9v = (3v)^2 + 3v = (3v + 9)(3v - 9)

However, this is not the correct factorization. We need to find another way to factor the polynomial.

Factoring the Polynomial Completely

After trying different methods, we find that the polynomial can be factored completely as:

9v2+9v−648=9(v+8)(v−9)9v^2 + 9v - 648 = 9(v + 8)(v - 9)

This is the correct factorization of the polynomial.

Conclusion

Factoring polynomials is a complex process that requires a deep understanding of algebraic concepts. In this article, we have explored different methods and techniques used to factor polynomials, and provided a step-by-step guide on how to factor the polynomial 9v2+9v−6489v^2 + 9v - 648 completely. We have also discussed the importance of checking the factored form to ensure that it is correct.

Final Answer

The final answer is:

9(v+8)(v−9)9(v + 8)(v - 9)

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Introduction

In our previous article, we explored the different methods and techniques used to factor polynomials, and provided a step-by-step guide on how to factor the polynomial 9v2+9v−6489v^2 + 9v - 648 completely. In this article, we will answer some of the most frequently asked questions about factoring polynomials.

Q&A

Q: What is factoring a polynomial?

A: Factoring a polynomial involves expressing it as a product of simpler polynomials. This can be done using various methods, including the factoring method, the grouping method, and the difference of squares method.

Q: What are the different types of factoring methods?

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

  • Factoring method: This involves finding two binomials whose product is equal to the given polynomial.
  • Grouping method: This involves grouping the terms of the polynomial into two pairs, and then factoring each pair separately.
  • Difference of squares method: This involves factoring a polynomial that can be written as the difference of two squares.

Q: How do I know which factoring method to use?

A: The choice of factoring method depends on the structure of the polynomial. For example, if the polynomial can be written as the difference of two squares, you should use the difference of squares method. If the polynomial has a common factor, you should use the factoring method.

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

A: Factoring a polynomial involves expressing it as a product of simpler polynomials, while simplifying a polynomial involves combining like terms to get a simpler expression.

Q: Can I factor a polynomial that has a variable in the denominator?

A: No, you cannot factor a polynomial that has a variable in the denominator. This is because the denominator cannot be factored in the same way as the numerator.

Q: How do I check if the factored form is correct?

A: To check if the factored form is correct, you can multiply the two binomials together and see if you get the original polynomial.

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

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

  • Not checking the factored form to ensure that it is correct.
  • Not using the correct factoring method for the given polynomial.
  • Not combining like terms to get a simpler expression.

Tips and Tricks

Tip 1: Use the factoring method when the polynomial has a common factor.

When the polynomial has a common factor, you can use the factoring method to factor it. This involves finding two binomials whose product is equal to the given polynomial.

Tip 2: Use the grouping method when the polynomial has a lot of terms.

When the polynomial has a lot of terms, you can use the grouping method to factor it. This involves grouping the terms of the polynomial into two pairs, and then factoring each pair separately.

Tip 3: Use the difference of squares method when the polynomial can be written as the difference of two squares.

When the polynomial can be written as the difference of two squares, you can use the difference of squares method to factor it. This involves factoring the polynomial as the difference of two squares.

Conclusion

Factoring polynomials is a complex process that requires a deep understanding of algebraic concepts. In this article, we have answered some of the most frequently asked questions about factoring polynomials, and provided some tips and tricks to help you factor polynomials more efficiently.

Final Answer

The final answer is:

  • Factoring a polynomial involves expressing it as a product of simpler polynomials.
  • The choice of factoring method depends on the structure of the polynomial.
  • You should check the factored form to ensure that it is correct.
  • You should avoid common mistakes such as not checking the factored form and not using the correct factoring method.