Factor Completely.$15t^3 + 5t^2 - 39t - 13$
Factor Completely: A Step-by-Step Guide to Factoring the Polynomial
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 completely. Factoring polynomials can be a challenging task, but with the right techniques and strategies, it can be done efficiently.
Before we begin factoring the polynomial, let's take a closer look at its structure. The polynomial is a cubic polynomial, meaning it has three terms. The first term, , is a cubic term, while the second term, , is a quadratic term. The third term, , is a linear term, and the fourth term, , is a constant term.
One of the most effective ways to factor a polynomial is by grouping its terms. Grouping involves dividing the polynomial into two or more groups of terms that can be factored separately. In this case, we can group the first two terms and the last two terms separately.
15t^3 + 5t^2 - 39t - 13 = (15t^3 + 5t^2) - (39t + 13)
Now that we have grouped the terms, let's focus on factoring out common factors from each group. In the first group, we can factor out a common factor of .
(15t^3 + 5t^2) = 5t^2(3t + 1)
In the second group, we can factor out a common factor of .
(39t + 13) = -13(3t + 1)
Now that we have factored out common factors from each group, let's focus on factoring the difference of squares. The difference of squares is a special case of factoring that involves expressing a polynomial as the difference of two squares.
a^2 - b^2 = (a + b)(a - b)
In this case, we can factor the difference of squares as follows:
5t^2(3t + 1) - 13(3t + 1) = (3t + 1)(5t^2 - 13)
Now that we have factored the difference of squares, let's focus on factoring the quadratic. The quadratic can be factored as follows:
5t^2 - 13 = (5t^2 - 25) + 12
Now that we have factored the quadratic, let's focus on factoring the difference of squares again. The difference of squares can be factored as follows:
5t^2 - 25 = (5t - 5)(t + 5)
Now that we have factored the difference of squares again, let's focus on factoring the final expression. The final expression can be factored as follows:
(5t - 5)(t + 5) + 12 = (5t - 5)(t + 5) + 4(3)
Now that we have factored the final expression, let's focus on factoring it again. The final expression can be factored as follows:
(5t - 5)(t + 5) + 4(3) = (5t - 5)(t + 5) + 12
In this article, we have factored the polynomial completely. We have used various techniques and strategies, including grouping terms, factoring out common factors, and factoring the difference of squares. By following these steps, we have been able to factor the polynomial into its simplest form.
The final answer is:
(5t - 5)(t + 5) + 12
Factoring polynomials is a fundamental concept in algebra that involves expressing a polynomial as a product of simpler polynomials. In this article, we will provide answers to some of the most frequently asked questions about factoring polynomials.
A: Factoring a polynomial involves expressing it as a product of simpler polynomials. This means that we can break down a polynomial into smaller parts, called factors, that can be multiplied together to get the original polynomial.
A: Factoring is important because it allows us to simplify polynomials and make them easier to work with. By factoring a polynomial, we can identify its roots and behavior, which is essential in many areas of mathematics and science.
A: There are several types of factoring, including:
- Factoring out common factors: This involves factoring out a common factor from each term in a polynomial.
- Factoring the difference of squares: This involves factoring a polynomial that can be expressed as the difference of two squares.
- Factoring the sum of squares: This involves factoring a polynomial that can be expressed as the sum of two squares.
- Factoring quadratic expressions: This involves factoring a quadratic expression into the product of two binomials.
A: Factoring a polynomial involves several steps, including:
- Grouping terms: Grouping involves dividing the polynomial into two or more groups of terms that can be factored separately.
- Factoring out common factors: Factoring out common factors involves factoring out a common factor from each term in a polynomial.
- Factoring the difference of squares: Factoring the difference of squares involves factoring a polynomial that can be expressed as the difference of two squares.
- Factoring the sum of squares: Factoring the sum of squares involves factoring a polynomial that can be expressed as the sum of two squares.
- Factoring quadratic expressions: Factoring quadratic expressions involves factoring a quadratic expression into the product of two binomials.
A: Some common mistakes to avoid when factoring polynomials include:
- Not grouping terms correctly: Grouping involves dividing the polynomial into two or more groups of terms that can be factored separately. If you don't group terms correctly, you may end up with incorrect factors.
- Not factoring out common factors correctly: Factoring out common factors involves factoring out a common factor from each term in a polynomial. If you don't factor out common factors correctly, you may end up with incorrect factors.
- Not recognizing the difference of squares: The difference of squares is a special case of factoring that involves expressing a polynomial as the difference of two squares. If you don't recognize the difference of squares, you may end up with incorrect factors.
- Not recognizing the sum of squares: The sum of squares is a special case of factoring that involves expressing a polynomial as the sum of two squares. If you don't recognize the sum of squares, you may end up with incorrect factors.
A: To check your work when factoring polynomials, you can use the following steps:
- Multiply the factors: Multiply the factors together to get the original polynomial.
- Check for errors: Check for errors in your work, such as incorrect factors or incorrect multiplication.
- Verify the result: Verify the result by checking that it is equal to the original polynomial.
Factoring polynomials is an essential skill in algebra that involves expressing a polynomial as a product of simpler polynomials. By following the steps outlined in this article, you can factor polynomials with ease and accuracy. Remember to group terms correctly, factor out common factors correctly, recognize the difference of squares, and recognize the sum of squares. By following these steps, you can ensure that your work is accurate and complete.