Completely Factor The Expression:${ 15x^2 + 7x - 2 }$A. { (3x + 2)(5x - 1)$}$ B. { (5x + 2)(3x - 1)$}$ C. { (15x + 1)(x - 2)$}$ D. { (15x - 1)(x + 2)$}$
Introduction
Factoring expressions is a fundamental concept in algebra, and it plays a crucial role in solving equations and inequalities. In this article, we will focus on completely factoring the expression 15x^2 + 7x - 2. Factoring an expression involves expressing it as a product of simpler expressions, called factors. The goal of factoring is to rewrite the expression in a way that makes it easier to solve or analyze.
Understanding the Expression
Before we start factoring, let's take a closer look at the expression 15x^2 + 7x - 2. This is a quadratic expression, which means it has a degree of 2. The expression consists of three terms: 15x^2, 7x, and -2. To factor this expression, we need to find two binomials whose product is equal to the given expression.
Factoring Techniques
There are several factoring techniques that we can use to factor the expression 15x^2 + 7x - 2. The most common techniques include:
- Factoring by Grouping: This involves grouping the terms of the expression into two groups and then factoring each group separately.
- Factoring by Greatest Common Factor (GCF): This involves finding the greatest common factor of the terms of the expression and factoring it out.
- Factoring Quadratics: This involves factoring quadratic expressions of the form ax^2 + bx + c.
Factoring by Grouping
Let's try factoring the expression 15x^2 + 7x - 2 using the factoring by grouping technique. We can group the terms of the expression into two groups: 15x^2 + 7x and -2.
15x^2 + 7x - 2
= (15x^2 + 7x) - 2
Factoring the First Group
Now, let's focus on factoring the first group: 15x^2 + 7x. We can factor out the greatest common factor of the terms in this group, which is 7x.
15x^2 + 7x
= 7x(15x + 1)
Factoring the Second Group
The second group is a constant term, -2. We can't factor this term further.
Putting it All Together
Now, let's put the factored form of the first group and the second group together to get the factored form of the entire expression.
15x^2 + 7x - 2
= (7x(15x + 1)) - 2
= 7x(15x + 1) - 2
Factoring the Entire Expression
To factor the entire expression, we need to find a way to combine the two groups into a single expression. We can do this by factoring out a common factor from both groups.
15x^2 + 7x - 2
= 7x(15x + 1) - 2
= 7x(15x + 1) - 2(1)
= 7x(15x + 1) - 2(1)
= (7x - 2)(15x + 1)
Conclusion
In this article, we completely factored the expression 15x^2 + 7x - 2. We used the factoring by grouping technique to factor the expression into two groups and then factored each group separately. Finally, we combined the factored form of the two groups into a single expression. The factored form of the expression is (7x - 2)(15x + 1).
Final Answer
The final answer is: (7x - 2)(15x + 1)
Introduction
In our previous article, we completely factored the expression 15x^2 + 7x - 2. We used the factoring by grouping technique to factor the expression into two groups and then factored each group separately. In this article, we will answer some common questions related to factoring expressions.
Q&A
Q: What is factoring in algebra?
A: Factoring in algebra involves expressing an expression as a product of simpler expressions, called factors. The goal of factoring is to rewrite the expression in a way that makes it easier to solve or analyze.
Q: What are the different factoring techniques?
A: There are several factoring techniques that we can use to factor expressions, including:
- Factoring by Grouping: This involves grouping the terms of the expression into two groups and then factoring each group separately.
- Factoring by Greatest Common Factor (GCF): This involves finding the greatest common factor of the terms of the expression and factoring it out.
- Factoring Quadratics: This involves factoring quadratic expressions of the form ax^2 + bx + c.
Q: How do I choose the right factoring technique?
A: To choose the right factoring technique, you need to analyze the expression and determine which technique is most suitable. For example, if the expression has a greatest common factor, you can use the factoring by GCF technique. If the expression is a quadratic, you can use the factoring quadratics technique.
Q: What is the difference between factoring and simplifying?
A: Factoring and simplifying are two different concepts in algebra. Factoring involves expressing an expression as a product of simpler expressions, while simplifying involves combining like terms to get a simpler expression.
Q: Can I factor an expression with a negative sign?
A: Yes, you can factor an expression with a negative sign. When factoring an expression with a negative sign, you need to consider the sign of each term and factor accordingly.
Q: How do I factor an expression with a variable in the denominator?
A: To factor an expression with a variable in the denominator, you need to multiply the numerator and denominator by the conjugate of the denominator. This will eliminate the variable in the denominator and allow you to factor the expression.
Q: Can I factor an expression with a fraction?
A: Yes, you can factor an expression with a fraction. When factoring an expression with a fraction, you need to consider the numerator and denominator separately and factor accordingly.
Q: How do I factor an expression with a coefficient?
A: To factor an expression with a coefficient, you need to consider the coefficient as a factor and factor the expression accordingly.
Q: What is the final answer to the expression 15x^2 + 7x - 2?
A: The final answer to the expression 15x^2 + 7x - 2 is (7x - 2)(15x + 1).
Conclusion
In this article, we answered some common questions related to factoring expressions. We discussed the different factoring techniques, how to choose the right technique, and how to factor expressions with negative signs, variables in the denominator, fractions, and coefficients. We also provided the final answer to the expression 15x^2 + 7x - 2.
Final Answer
The final answer is: (7x - 2)(15x + 1)