Understanding the X Method

Oct 13, 2025 by windailys.com 91 views

Introduction

Factoring polynomials is an essential skill in algebra, and it can be a bit challenging, especially when dealing with quadratic expressions. In this article, we will focus on factoring the polynomial $x^2 + 13x - 48$ completely using the X method. We will also explore the four-term polynomial and factored form of the polynomial.

Understanding the X Method

The X method is a technique used to factor quadratic expressions of the form $ax^2 + bx + c$ . It involves finding two numbers whose product is $ac$ and whose sum is $b$ . These numbers are the factors of the quadratic expression.

Step 1: Find the Factors of the Constant Term

The constant term of the polynomial is $-48$ . We need to find two numbers whose product is $-48$ . The factors of $-48$ are:

$1$ and $-48$
$2$ and $-24$
$3$ and $-16$
$4$ and $-12$
$6$ and $-8$
$8$ and $-6$
$12$ and $-4$
$16$ and $-3$
$24$ and $-2$
$48$ and $-1$

Step 2: Find the Factors of the Coefficient of the x Term

The coefficient of the $x$ term is $13$ . We need to find two numbers whose sum is $13$ . The pairs of numbers whose sum is $13$ are:

$1$ and $12$
$2$ and $11$
$3$ and $10$
$4$ and $9$
$5$ and $8$
$6$ and $7$

Step 3: Combine the Factors

We need to find the combination of factors that satisfies both conditions: the product of the factors is $-48$ and the sum of the factors is $13$ . After analyzing the factors, we find that the combination of factors that satisfies both conditions is:

$-1$ and $-48$
$-2$ and $-24$
$-3$ and $-16$
$-4$ and $-12$
$-6$ and $-8$
$-8$ and $-6$
$-12$ and $-4$
$-16$ and $-3$
$-24$ and $-2$
$-48$ and $-1$

However, we are looking for a combination of factors that satisfies both conditions. After analyzing the factors, we find that the combination of factors that satisfies both conditions is:

$-1$ and $-48$ (sum is $-49$ , not $13$ )
$-2$ and $-24$ (sum is $-26$ , not $13$ )
$-3$ and $-16$ (sum is $-19$ , not $13$ )
$-4$ and $-12$ (sum is $-16$ , not $13$ )
$-6$ and $-8$ (sum is $-14$ , not $13$ )
$-8$ and $-6$ (sum is $-14$ , not $13$ )
$-12$ and $-4$ (sum is $-16$ , not $13$ )
$-16$ and $-3$ (sum is $-19$ , not $13$ )
$-24$ and $-2$ (sum is $-26$ , not $13$ )
$-48$ and $-1$ (sum is $-49$ , not $13$ )

However, we are looking for a combination of factors that satisfies both conditions. After analyzing the factors, we find that the combination of factors that satisfies both conditions is:

$6$ and $-8$ (product is $-48$ , sum is $-2$ , not $13$ )
$8$ and $-6$ (product is $-48$ , sum is $2$ , not $13$ )
$12$ and $-4$ (product is $-48$ , sum is $8$ , not $13$ )
$16$ and $-3$ (product is $-48$ , sum is $13$ , this is the correct combination)
$24$ and $-2$ (product is $-48$ , sum is $22$ , not $13$ )
$48$ and $-1$ (product is $-48$ , sum is $47$ , not $13$ )

Step 4: Write the Factored Form

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

$x^2 + 13x - 48 = (x + 16)(x - 3)$

Conclusion

In this article, we have used the X method to factor the polynomial $x^2 + 13x - 48$ completely. We have found the four-term polynomial and factored form of the polynomial, which is $(x + 16)(x - 3)$ . This is a useful technique for factoring quadratic expressions and can be applied to a wide range of problems.

Final Answer

$The final answer is: $\boxed{(x + 16)(x - 3)}$<br/> **Q&A: Factoring Polynomials Using the X Method**$

Introduction

In our previous article, we explored the X method for factoring polynomials. This technique is a powerful tool for simplifying quadratic expressions and can be applied to a wide range of problems. In this article, we will answer some common questions about factoring polynomials using the X method.

Q: What is the X method?

A: The X method is a technique used to factor quadratic expressions of the form $ax^2 + bx + c$ . It involves finding two numbers whose product is $ac$ and whose sum is $b$ . These numbers are the factors of the quadratic expression.

Q: How do I find the factors of the constant term?

A: To find the factors of the constant term, you need to list all the possible pairs of numbers whose product is equal to the constant term. For example, if the constant term is $-48$ , you would list all the possible pairs of numbers whose product is $-48$ .

Q: How do I find the factors of the coefficient of the x term?

A: To find the factors of the coefficient of the x term, you need to list all the possible pairs of numbers whose sum is equal to the coefficient of the x term. For example, if the coefficient of the x term is $13$ , you would list all the possible pairs of numbers whose sum is $13$ .

Q: How do I combine the factors?

A: To combine the factors, you need to find the pair of factors that satisfies both conditions: the product of the factors is equal to the constant term and the sum of the factors is equal to the coefficient of the x term.

Q: What if I don't find a pair of factors that satisfies both conditions?

A: If you don't find a pair of factors that satisfies both conditions, it may be because the quadratic expression cannot be factored using the X method. In this case, you may need to use other factoring techniques or seek help from a teacher or tutor.

Q: Can I use the X method to factor any quadratic expression?

A: No, the X method can only be used to factor quadratic expressions of the form $ax^2 + bx + c$ where $a$ , $b$ , and $c$ are integers. If the coefficients of the quadratic expression are not integers, you may need to use other factoring techniques.

Q: How do I write the factored form of the polynomial?

A: To write the factored form of the polynomial, you need to multiply the two factors together. For example, if the factored form of the polynomial is $(x + 16)(x - 3)$ , you would multiply the two factors together to get $x^2 + 13x - 48$ .

Q: What are some common mistakes to avoid when using the X method?

A: Some common mistakes to avoid when using the X method include:

Not listing all the possible pairs of numbers whose product is equal to the constant term
Not listing all the possible pairs of numbers whose sum is equal to the coefficient of the x term
Not finding the pair of factors that satisfies both conditions
Not multiplying the two factors together to get the factored form of the polynomial

Conclusion

In this article, we have answered some common questions about factoring polynomials using the X method. We hope that this information has been helpful in understanding this technique and how to apply it to a wide range of problems.

Final Answer

The final answer is: $\boxed{(x + 16)(x - 3)}$