If A Polynomial Has Four Terms, $3x^3 + 5x + 6x^2 + 10$, Which Factoring Method Can Be Considered?A. Perfect-square Trinomial B. Difference Of Squares C. Factor By Grouping D. Sum Of Cubes

Introduction

Polynomials are a fundamental concept in algebra, and factoring them is a crucial skill for students and professionals alike. In this article, we will explore the different methods of factoring polynomials, with a focus on the specific case of a four-term polynomial. We will examine the options available for factoring the polynomial $3x^3 + 5x + 6x^2 + 10$ and determine which method is most suitable.

Understanding the Polynomial

Before we dive into the factoring methods, let's take a closer look at the polynomial $3x^3 + 5x + 6x^2 + 10$. This polynomial has four terms, with the highest degree term being $3x^3$. The other terms are $5x$, $6x^2$, and a constant term of $10$.

Factoring Methods

There are several methods for factoring polynomials, each with its own strengths and weaknesses. Let's examine the options available for factoring the polynomial $3x^3 + 5x + 6x^2 + 10$.

A. Perfect-Square Trinomial

A perfect-square trinomial is a polynomial that can be factored into the square of a binomial. The general form of a perfect-square trinomial is $a^2 + 2ab + b^2 = (a + b)^2$. To determine if a polynomial is a perfect-square trinomial, we need to check if it can be written in this form.

In the case of the polynomial $3x^3 + 5x + 6x^2 + 10$, we can see that it does not fit the form of a perfect-square trinomial. The polynomial has a cubic term, which is not present in a perfect-square trinomial.

B. Difference of Squares

A difference of squares is a polynomial that can be factored into the difference of two squares. The general form of a difference of squares is $a^2 - b^2 = (a + b)(a - b)$. To determine if a polynomial is a difference of squares, we need to check if it can be written in this form.

In the case of the polynomial $3x^3 + 5x + 6x^2 + 10$, we can see that it does not fit the form of a difference of squares. The polynomial has a cubic term and a constant term, which are not present in a difference of squares.

C. Factor by Grouping

Factor by grouping is a method of factoring polynomials that involves grouping terms together and factoring out common factors. This method is particularly useful for polynomials with multiple terms.

In the case of the polynomial $3x^3 + 5x + 6x^2 + 10$, we can try factoring by grouping. We can group the terms $3x^3$ and $6x^2$ together, and the terms $5x$ and $10$ together.

(3x^3 + 6x^2) + (5x + 10)

We can then factor out a common factor from each group.

3x^2(x + 2) + 5(x + 2)

We can see that both groups have a common factor of $(x + 2)$. We can factor this out to get:

(3x^2 + 5)(x + 2)

This is the factored form of the polynomial $3x^3 + 5x + 6x^2 + 10$.

D. Sum of Cubes

A sum of cubes is a polynomial that can be factored into the sum of two cubes. The general form of a sum of cubes is $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$. To determine if a polynomial is a sum of cubes, we need to check if it can be written in this form.

In the case of the polynomial $3x^3 + 5x + 6x^2 + 10$, we can see that it does not fit the form of a sum of cubes. The polynomial has a constant term, which is not present in a sum of cubes.

Conclusion

In conclusion, the most suitable method for factoring the polynomial $3x^3 + 5x + 6x^2 + 10$ is factor by grouping. This method involves grouping terms together and factoring out common factors. By using this method, we can factor the polynomial into the form $(3x^2 + 5)(x + 2)$.

Final Answer

The final answer is C. Factor by grouping.