Factoring By Grouping How To Factor X^3-7x^2-5x+35

Factoring polynomials is a fundamental skill in algebra, and one powerful technique for doing so is factoring by grouping. This method is particularly useful when dealing with polynomials that have four or more terms. In this article, we will walk through the process of factoring the cubic polynomial $x^3 - 7x^2 - 5x + 35$ by grouping, providing a step-by-step explanation to ensure clarity and understanding. This polynomial expression might seem complex at first, but by using the factoring by grouping method, we can break it down into simpler, more manageable parts. The goal is to rewrite the polynomial as a product of lower-degree polynomials, which makes it easier to solve equations and analyze functions. Let's dive into the details and explore how this technique works.

Understanding Factoring by Grouping

Factoring by grouping is a technique used to factor polynomials with four or more terms. The basic idea is to group terms in pairs, factor out the greatest common factor (GCF) from each pair, and then look for a common binomial factor. This method relies on the distributive property in reverse, allowing us to rewrite a polynomial as a product of simpler expressions. Before we delve into our specific example, let's understand the underlying principle. Suppose we have a polynomial expression like $ax + ay + bx + by$. Notice that we can group the first two terms and the last two terms: $(ax + ay) + (bx + by)$. From the first group, we can factor out $a$, and from the second group, we can factor out $b$. This gives us $a(x + y) + b(x + y)$. Now, we observe that $(x + y)$ is a common factor in both terms. We can factor it out, resulting in $(x + y)(a + b)$. This simple example illustrates the core idea behind factoring by grouping: identify common factors within groups of terms and then factor out the common binomial. Now, let’s apply this method to our given polynomial $x^3 - 7x^2 - 5x + 35$.

Step-by-Step Factoring of $x^3 - 7x^2 - 5x + 35$

Step 1: Group the Terms

The first step in factoring by grouping is to group the terms in pairs. We look for terms that might have a common factor. In the polynomial $x^3 - 7x^2 - 5x + 35$, a natural grouping is to pair the first two terms and the last two terms together. This gives us: $(x^3 - 7x^2) + (-5x + 35)$. Grouping terms like this allows us to focus on smaller parts of the expression and identify potential common factors more easily. It's crucial to maintain the signs correctly when grouping; the negative sign in front of $5x$ remains with the term when grouped. This initial grouping sets the stage for the next steps, where we'll factor out the greatest common factor from each group.

Step 2: Factor out the Greatest Common Factor (GCF) from Each Group

Now that we have grouped the terms, we need to factor out the greatest common factor (GCF) from each group. For the first group, $(x^3 - 7x^2)$, the GCF is $x^2$. Factoring $x^2$ from both terms gives us $x^2(x - 7)$. For the second group, $(-5x + 35)$, the GCF is $-5$. Factoring $-5$ from both terms gives us $-5(x - 7)$. Notice that by factoring out $-5$ instead of $5$, we obtain the same binomial factor $(x - 7)$ as in the first group. This is a crucial step in factoring by grouping because having the same binomial factor in both groups allows us to proceed with the next step. So, after factoring out the GCF from each group, our expression looks like this: $x^2(x - 7) - 5(x - 7)$.

Step 3: Factor out the Common Binomial

At this stage, we have the expression $x^2(x - 7) - 5(x - 7)$. Notice that $(x - 7)$ is a common binomial factor in both terms. We can factor out this common binomial, just as we would factor out a single term. Factoring $(x - 7)$ from the entire expression gives us $(x - 7)(x^2 - 5)$. This is the result of factoring by grouping. We have successfully rewritten the original cubic polynomial as a product of two lower-degree polynomials: a linear factor $(x - 7)$ and a quadratic factor $(x^2 - 5)$. This factored form is much easier to work with when solving equations or analyzing the polynomial's behavior. The process of factoring out the common binomial is the key step in this method, as it transforms the expression into a product of factors.

Step 4: Check if Further Factoring is Possible

After factoring by grouping, it's always a good practice to check if further factoring is possible. In our case, we have the expression $(x - 7)(x^2 - 5)$. The linear factor $(x - 7)$ cannot be factored further. However, we need to examine the quadratic factor $(x^2 - 5)$. This expression is a difference of squares in the form $a^2 - b^2$, where $a = x$ and $b = \sqrt{5}$. The difference of squares can be factored as $(a - b)(a + b)$. Applying this to our quadratic factor, we get $(x - \sqrt{5})(x + \sqrt{5})$. Therefore, the fully factored form of the polynomial is $(x - 7)(x - \sqrt{5})(x + \sqrt{5})$. However, if we are looking for factors with integer coefficients, we stop at $(x - 7)(x^2 - 5)$ since the square root of 5 is not an integer. In many contexts, factoring by grouping is sufficient to simplify the polynomial, and the form $(x - 7)(x^2 - 5)$ is considered the final factored form. Always consider the context of the problem to determine if further factoring is necessary.

Final Result and Conclusion

After applying the factoring by grouping method to the polynomial $x^3 - 7x^2 - 5x + 35$, we arrive at the factored form $(x^2 - 5)(x - 7)$. Therefore, the correct answer is $(x^2 - 5)(x - 7)$. This process illustrates the power of factoring by grouping in simplifying complex polynomial expressions. By grouping terms, factoring out common factors, and identifying common binomials, we can rewrite polynomials in a more manageable form. This skill is essential for solving polynomial equations, simplifying algebraic expressions, and understanding the behavior of polynomial functions. Factoring allows us to break down complex expressions into simpler components, making them easier to analyze and manipulate. Always remember to check for further factoring possibilities, such as the difference of squares, to ensure the polynomial is fully factored.

In conclusion, factoring by grouping is a valuable technique in algebra that enables us to factor polynomials with four or more terms. By following a systematic approach—grouping terms, factoring out the GCF from each group, factoring out the common binomial, and checking for further factoring—we can simplify polynomial expressions and make them easier to work with. This method is a cornerstone of algebraic manipulation and is crucial for solving a wide range of mathematical problems. Whether you're solving equations, simplifying expressions, or analyzing functions, mastering factoring by grouping will undoubtedly enhance your algebraic skills and problem-solving abilities.

Answer:

\(x^2-5)(x-7)