Factoring polynomials with rational coefficients

Oct 13, 2025 by miodeya.com 115 views

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 quadratic polynomials, which are polynomials of the form $ax^2 + bx + c$ . We will use the given polynomial $x^2 + 15x + 56$ as an example to demonstrate the factoring process.

The Factoring Process

To factor a quadratic polynomial, we need to find two numbers whose product is equal to the constant term (in this case, 56) and whose sum is equal to the coefficient of the linear term (in this case, 15). These two numbers are the roots of the polynomial, and they can be used to write the polynomial as a product of two binomials.

Step 1: Find the Factors of the Constant Term

The first step in factoring the polynomial is to find the factors of the constant term, which is 56. We can start by listing the factors of 56:

1 and 56
2 and 28
4 and 14
7 and 8

Step 2: Find the Pair of Factors that Add Up to the Coefficient of the Linear Term

Now that we have listed the factors of 56, we need to find the pair of factors that add up to the coefficient of the linear term, which is 15. We can start by checking each pair of factors:

1 + 56 = 57 (not equal to 15)
2 + 28 = 30 (not equal to 15)
4 + 14 = 18 (not equal to 15)
7 + 8 = 15 (this is the correct pair of factors)

Step 3: Write the Polynomial as a Product of Two Binomials

Now that we have found the pair of factors that add up to 15, we can write the polynomial as a product of two binomials:

$x^2 + 15x + 56 = (x + 7)(x + 8)$

Conclusion

In this article, we have demonstrated the factoring process for the polynomial $x^2 + 15x + 56$ . We have shown that the polynomial can be factored as $(x + 7)(x + 8)$ . This is just one example of how to factor a quadratic polynomial, and there are many other techniques and strategies that can be used to factor polynomials.

Answer

Based on the factoring process we have demonstrated, the correct answer is:

(A) $(x+8)(x+7)$

Other Options

The other options are not correct because they do not factor the polynomial correctly. Option B is incorrect because it does not factor the polynomial correctly. Option C is incorrect because it has a negative sign in the first binomial, which is not consistent with the factoring process we have demonstrated. Option D is incorrect because it has a negative sign in both binomials, which is not consistent with the factoring process we have demonstrated.

Discussion

Factoring polynomials is an important concept in algebra that has many real-world applications. In this article, we have demonstrated the factoring process for a quadratic polynomial, and we have shown that the polynomial can be factored as $(x + 7)(x + 8)$ . This is just one example of how to factor a quadratic polynomial, and there are many other techniques and strategies that can be used to factor polynomials.

Additional Resources

For more information on factoring polynomials, please see the following resources:

Khan Academy: Factoring Quadratic Expressions
Mathway: Factoring Polynomials
Wolfram Alpha: Factoring Polynomials

Conclusion

Introduction

Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In our previous article, we demonstrated the factoring process for a quadratic polynomial, and we showed that the polynomial can be factored as $(x + 7)(x + 8)$ . In this article, we will answer some frequently asked questions about factoring polynomials.

Q: What is factoring a polynomial?

A: Factoring a polynomial involves expressing it as a product of simpler polynomials. For example, the polynomial $x^2 + 15x + 56$ can be factored as $(x + 7)(x + 8)$ .

Q: Why is factoring a polynomial important?

A: Factoring a polynomial is important because it allows us to simplify complex expressions and solve equations more easily. It also helps us to identify the roots of a polynomial, which is essential in many real-world applications.

Q: What are the different types of factoring?

A: There are several types of factoring, including:

Factoring out a greatest common factor (GCF): This involves factoring out the largest common factor of all the terms in the polynomial.
Factoring by grouping: This involves grouping the terms in the polynomial into pairs and factoring each pair separately.
Factoring quadratics: This involves factoring quadratic polynomials, such as $x^2 + 15x + 56$ .
Factoring polynomials with rational coefficients: This involves factoring polynomials with rational coefficients, such as $x^2 + \frac{1}{2}x + \frac{1}{4}$ .

Q: How do I factor a polynomial?

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

Check if the polynomial can be factored by grouping: If the polynomial can be grouped into pairs, try factoring each pair separately.
Check if the polynomial can be factored by factoring out a GCF: If the polynomial has a common factor, try factoring it out.
Check if the polynomial can be factored as a quadratic: If the polynomial is a quadratic, try factoring it as a quadratic.
Use the quadratic formula: If the polynomial is a quadratic and cannot be factored, use the quadratic formula to find the roots.

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

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

Not checking if the polynomial can be factored by grouping: Make sure to check if the polynomial can be grouped into pairs before trying to factor it.
Not checking if the polynomial can be factored by factoring out a GCF: Make sure to check if the polynomial has a common factor before trying to factor it.
Not using the quadratic formula: If the polynomial is a quadratic and cannot be factored, make sure to use the quadratic formula to find the roots.

Q: How do I know if a polynomial can be factored?

A: To determine if a polynomial can be factored, you can try the following:

Check if the polynomial has a common factor: If the polynomial has a common factor, it can be factored by factoring out the GCF.
Check if the polynomial can be grouped into pairs: If the polynomial can be grouped into pairs, it can be factored by grouping.
Check if the polynomial is a quadratic: If the polynomial is a quadratic, it can be factored as a quadratic.

Conclusion

In conclusion, factoring polynomials is an important concept in algebra that has many real-world applications. We have answered some frequently asked questions about factoring polynomials, and we have provided some tips and tricks for factoring polynomials. By following these tips and tricks, you can become a master of factoring polynomials and solve equations more easily.