When Asked To Factor The Trinomial $x^2 - 18x + 81$, A Student Gives The Answer $(x - 9)(x + 9)$. Which Of The Following Statements Is True?A. The Answer Is Incorrect; The Plus Sign Should Be A Minus Sign.B. The Answer Is

Introduction

Factoring trinomials is a fundamental concept in algebra, and it requires a deep understanding of the underlying mathematical principles. When a student is asked to factor a trinomial, they must carefully examine the expression and determine the correct factors. In this article, we will explore a common scenario where a student is asked to factor the trinomial $x^2 - 18x + 81$ and provides the answer $(x - 9)(x + 9)$. We will examine the correctness of this answer and determine which of the given statements is true.

The Trinomial to be Factored

The trinomial to be factored is $x^2 - 18x + 81$. This expression can be factored using the method of factoring by grouping or by using the formula for factoring a quadratic expression of the form $ax^2 + bx + c$. In this case, we will use the formula for factoring a quadratic expression.

The Formula for Factoring a Quadratic Expression

The formula for factoring a quadratic expression of the form $ax^2 + bx + c$ is given by:

$$a(x - r_1)(x - r_2)$$

where $r_1$ and $r_2$ are the roots of the quadratic equation $ax^2 + bx + c = 0$.

Finding the Roots of the Quadratic Equation

To find the roots of the quadratic equation $x^2 - 18x + 81 = 0$, we can use the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, $a = 1$, $b = -18$, and $c = 81$. Plugging these values into the quadratic formula, we get:

$$x = \frac{-(-18) \pm \sqrt{(-18)^2 - 4(1)(81)}}{2(1)}$$

Simplifying the expression, we get:

$$x = \frac{18 \pm \sqrt{324 - 324}}{2}$$

$$x = \frac{18 \pm \sqrt{0}}{2}$$

$$x = \frac{18}{2}$$

$$x = 9$$

Therefore, the roots of the quadratic equation are $x = 9$ and $x = 9$.

Factoring the Trinomial

Since the roots of the quadratic equation are $x = 9$ and $x = 9$, we can factor the trinomial as follows:

$$x^2 - 18x + 81 = (x - 9)(x - 9)$$

$$x^2 - 18x + 81 = (x - 9)^2$$

Therefore, the correct factorization of the trinomial is $(x - 9)^2$.

Conclusion

In conclusion, the student's answer $(x - 9)(x + 9)$ is incorrect. The correct factorization of the trinomial is $(x - 9)^2$. Therefore, the correct statement is:

A. The answer is incorrect; the plus sign should be a minus sign.

This statement is true because the student's answer does not match the correct factorization of the trinomial.

Discussion

This scenario highlights the importance of carefully examining the expression to be factored and determining the correct factors. It also emphasizes the need to use the correct formula for factoring a quadratic expression and to carefully simplify the expression.

Common Mistakes

When factoring trinomials, students often make common mistakes, such as:

• Not carefully examining the expression to be factored
• Not using the correct formula for factoring a quadratic expression
• Not carefully simplifying the expression
• Not checking the answer for correctness

Tips for Factoring Trinomials

To avoid making common mistakes when factoring trinomials, students should:

• Carefully examine the expression to be factored
• Use the correct formula for factoring a quadratic expression
• Carefully simplify the expression
• Check the answer for correctness

By following these tips, students can ensure that they provide the correct factorization of a trinomial.

Conclusion

$$$$$$

Introduction

In our previous article, we explored a common scenario where a student is asked to factor the trinomial $x^2 - 18x + 81$ and provided the answer $(x - 9)(x + 9)$. We determined that the student's answer is incorrect and that the correct factorization of the trinomial is $(x - 9)^2$. In this article, we will provide a Q&A section to help students and teachers understand the concept of factoring trinomials and to address common questions and concerns.

Q&A

Q: What is the correct factorization of the trinomial $x^2 - 18x + 81$?

A: The correct factorization of the trinomial $x^2 - 18x + 81$ is $(x - 9)^2$.

Q: Why is the student's answer $(x - 9)(x + 9)$ incorrect?

A: The student's answer $(x - 9)(x + 9)$ is incorrect because it does not match the correct factorization of the trinomial. The correct factorization is $(x - 9)^2$, which is a perfect square trinomial.

Q: What is a perfect square trinomial?

A: A perfect square trinomial is a trinomial that can be factored into the square of a binomial. In this case, the trinomial $x^2 - 18x + 81$ is a perfect square trinomial because it can be factored into $(x - 9)^2$.

Q: How do I determine if a trinomial is a perfect square trinomial?

A: To determine if a trinomial is a perfect square trinomial, you can use the following steps:

1. Check if the first and last terms are perfect squares.
2. Check if the middle term is twice the product of the square roots of the first and last terms.
3. If the trinomial meets these conditions, it is a perfect square trinomial.

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

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

• Not carefully examining the expression to be factored
• Not using the correct formula for factoring a quadratic expression
• Not carefully simplifying the expression
• Not checking the answer for correctness

Q: How can I practice factoring trinomials?

A: You can practice factoring trinomials by:

• Working through example problems in your textbook or online resources
• Creating your own practice problems
• Using online tools or software to generate practice problems
• Working with a study group or tutor to practice factoring trinomials

Q: What are some tips for factoring trinomials?

A: Some tips for factoring trinomials include:

• Carefully examining the expression to be factored
• Using the correct formula for factoring a quadratic expression
• Carefully simplifying the expression
• Checking the answer for correctness
• Practicing regularly to build your skills and confidence

Conclusion

In conclusion, factoring trinomials is a fundamental concept in algebra that requires a deep understanding of the underlying mathematical principles. By following the correct steps and avoiding common mistakes, students can ensure that they provide the correct factorization of a trinomial. We hope that this Q&A section has been helpful in addressing common questions and concerns and in providing tips and resources for practicing factoring trinomials.