Factor The Polynomial: 3 X 3 + X 2 − 75 X − 25 3x^3 + X^2 - 75x - 25 3 X 3 + X 2 − 75 X − 25 Drag The Expressions Into The Box If They Are Part Of The Factored Form Of The Polynomial.- ( X 2 − 25 (x^2 - 25 ( X 2 − 25 ]- ( 3 X − 1 (3x - 1 ( 3 X − 1 ]- ( X − 5 (x - 5 ( X − 5 ]- ( X + 5 (x + 5 ( X + 5 ]- ( 3 X + 1 (3x + 1 ( 3 X + 1 ]-

Mar 8, 2025 by ADMIN 334 views

**Factoring the Polynomial: A Step-by-Step Guide**

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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 $3x^3 + x^2 - 75x - 25$ . We will use various factoring techniques to break down the polynomial into its factored form.

Understanding the Polynomial

Before we begin factoring, let's take a closer look at the given polynomial: $3x^3 + x^2 - 75x - 25$ . This polynomial is a cubic polynomial, meaning it has a degree of 3. The polynomial has four terms: $3x^3$ , $x^2$ , $-75x$ , and $-25$ .

Factoring Techniques

There are several factoring techniques that we can use to factor the polynomial. Some of the most common techniques include:

Greatest Common Factor (GCF): This technique involves finding the greatest common factor of all the terms in the polynomial.
Factoring by Grouping: This technique involves grouping the terms in the polynomial into pairs and factoring out the greatest common factor from each pair.
Factoring Quadratics: This technique involves factoring quadratic expressions of the form $ax^2 + bx + c$ .
Factoring by Synthetic Division: This technique involves using synthetic division to factor polynomials.

Factoring the Polynomial

Let's start by factoring the polynomial using the greatest common factor technique. We can see that the greatest common factor of all the terms is 1, so we cannot factor out any common factors.

Next, let's try factoring by grouping. We can group the first two terms and the last two terms:

$3x^3 + x^2 - 75x - 25 = (3x^3 + x^2) - (75x + 25)$

Now, we can factor out the greatest common factor from each pair:

$(3x^3 + x^2) - (75x + 25) = x^2(3x + 1) - 25(3x + 1)$

We can see that both terms have a common factor of $(3x + 1)$ , so we can factor it out:

$x^2(3x + 1) - 25(3x + 1) = (3x + 1)(x^2 - 25)$

Now, we can factor the quadratic expression $x^2 - 25$ using the difference of squares formula:

$x^2 - 25 = (x - 5)(x + 5)$

Therefore, the factored form of the polynomial is:

$(3x + 1)(x^2 - 25) = (3x + 1)(x - 5)(x + 5)$

Conclusion

In this article, we factored the polynomial $3x^3 + x^2 - 75x - 25$ using various factoring techniques. We started by factoring out the greatest common factor, then used factoring by grouping to factor out the common factor $(3x + 1)$ . Finally, we used the difference of squares formula to factor the quadratic expression $x^2 - 25$ . The factored form of the polynomial is $(3x + 1)(x - 5)(x + 5)$ .

Drag the Expressions into the Box

If the expressions are part of the factored form of the polynomial, drag them into the box:

$(x^2 - 25)$ : Drag into the box
$(3x - 1)$ : Drag into the box
$(x - 5)$ : Drag into the box
$(x + 5)$ : Drag into the box
$(3x + 1)$ : Drag into the box

Note: The correct answer is $(3x + 1)(x - 5)(x + 5)$ .

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Introduction

In our previous article, we factored the polynomial $3x^3 + x^2 - 75x - 25$ using various factoring techniques. In this article, we will provide a Q&A guide to help you understand the concept of factoring polynomials and how to apply it to different types of polynomials.

Q&A

Q: What is factoring a polynomial?

A: Factoring a polynomial involves expressing it as a product of simpler polynomials. This is done by finding the greatest common factor of all the terms in the polynomial and factoring it out.

Q: What are the different factoring techniques?

A: There are several factoring techniques that can be used to factor polynomials, including:

Greatest Common Factor (GCF): This technique involves finding the greatest common factor of all the terms in the polynomial.
Factoring by Grouping: This technique involves grouping the terms in the polynomial into pairs and factoring out the greatest common factor from each pair.
Factoring Quadratics: This technique involves factoring quadratic expressions of the form $ax^2 + bx + c$ .
Factoring by Synthetic Division: This technique involves using synthetic division to factor polynomials.

Q: How do I factor a polynomial using the greatest common factor technique?

A: To factor a polynomial using the greatest common factor technique, follow these steps:

Identify the greatest common factor of all the terms in the polynomial.
Factor out the greatest common factor from each term.
Write the factored form of the polynomial.

Q: How do I factor a polynomial using factoring by grouping?

A: To factor a polynomial using factoring by grouping, follow these steps:

Group the terms in the polynomial into pairs.
Factor out the greatest common factor from each pair.
Write the factored form of the polynomial.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, follow these steps:

Identify the coefficients of the quadratic expression.
Use the quadratic formula to find the roots of the quadratic expression.
Write the factored form of the quadratic expression.

Q: What is the difference of squares formula?

A: The difference of squares formula is a formula that can be used to factor quadratic expressions of the form $x^2 - a^2$ . The formula is:

$x^2 - a^2 = (x - a)(x + a)$

Q: How do I use the difference of squares formula to factor a polynomial?

A: To use the difference of squares formula to factor a polynomial, follow these steps:

Identify the quadratic expression in the polynomial.
Use the difference of squares formula to factor the quadratic expression.
Write the factored form of the polynomial.

Conclusion

In this article, we provided a Q&A guide to help you understand the concept of factoring polynomials and how to apply it to different types of polynomials. We covered various factoring techniques, including the greatest common factor technique, factoring by grouping, factoring quadratics, and factoring by synthetic division. We also covered the difference of squares formula and how to use it to factor quadratic expressions.

Common Mistakes to Avoid

When factoring polynomials, there are several common mistakes to avoid. These include:

Not identifying the greatest common factor: Make sure to identify the greatest common factor of all the terms in the polynomial before factoring it out.
Not grouping the terms correctly: Make sure to group the terms in the polynomial into pairs before factoring out the greatest common factor.
Not using the correct factoring technique: Make sure to use the correct factoring technique for the type of polynomial you are working with.
Not checking your work: Make sure to check your work to ensure that the factored form of the polynomial is correct.

Practice Problems

To practice factoring polynomials, try the following problems:

Factor the polynomial $2x^3 + 3x^2 - 4x - 6$ .
Factor the polynomial $x^2 + 5x + 6$ .
Factor the polynomial $x^3 - 2x^2 - 5x + 10$ .

Note: The answers to these problems can be found in the next article.