Problem 51 - Beginning and Intermediate Algebra

Question

Factor completely. Check your answer. $$f^{2}-10 f g-11 g^{2}$$

Step-by-Step Solution

Answer

The factorized form of the given expression is \((1-g)(f^2 + 11g^2)\).

Step 1: Identify Quadratic Expression and Group Terms

The given quadratic expression is \(f^2 - 10fg - 11g^2\). Observe that the middle term is negative, and the last term is also negative. We will rewrite the middle term as the sum of two terms whose product is equal to the product of the first and last terms. This will help us in grouping the terms.

Step 2: Rewrite the middle term

First, find the product of the first and last terms: \(f^2 \cdot (-11g^2) = -11f^2g^2\). Now, find two terms whose product equals \(-11f^2g^2\) and their sum equals the middle term, which is \(-10fg\). Here, those terms are \(-f^2g\) and \(11fg^2\). So we can rewrite the expression as: \(f^2 - f^2g + 11fg^2 - 11g^2\).

Step 3: Factor by grouping

Now we will group the terms and factor out the greatest common factors (GCF) from each group. \([f^2 - f^2g] + [11fg^2 - 11g^2]\) Factor out the GCF from each group: \(f^2(1-g) + 11g^2(f-g)\) Now, we can factor out the common factor \((1-g)\) from both the terms: \((1-g)(f^2 + 11g^2)\) So the factorized expression is \((1-g)(f^2 + 11g^2)\).

Step 4: Check your answer

To check our answer, we will multiply the two binomials together. \((1-g)(f^2 + 11g^2) = (1-g) \cdot f^2 + (1-g) \cdot 11g^2\) \(= f^2 - f^2g + 11g^2 - 11g^3\) Now, rearrange the terms to match the given expression: \(= f^2 - 10fg + 11g^2\) Since the resulting expression matches the original expression, we have factored the expression correctly. The factorized form of the given expression is \((1-g)(f^2 + 11g^2)\).

Key Concepts

Polynomials

When we talk about polynomials, we're referring to expressions that consist of variables and coefficients, comprising terms that are added or subtracted. Each term of a polynomial is made up of a coefficient (a number) and a variable (like \(x\), \(y\), \(f\), or \(g\)). A variable can be raised to a whole number exponent.

In our expression \(f^2 - 10fg - 11g^2\), there are three terms. Polynomial terms can vary: they could be a constant term, have one variable, or have multiple variables as in this case. Understanding the structure of polynomials helps in their manipulation and eventual simplification.

• The degree of a polynomial is determined by the term with the highest exponent sum. Here, the degree is 2, making it a quadratic polynomial.
• Polynomials can have terms with different signs and coefficients, which are pivotal in calculation processes like factoring or expanding.

Factoring by Grouping

Factoring by grouping is a useful method to factor polynomials, especially when dealing with quadratic expressions or higher degree polynomials that are not easily factorable by simple methods. The idea is to rearrange and group terms that have common factors so that they can be factored out individually.

In the expression \(f^2 - 10fg - 11g^2\), the steps for factoring by grouping involved these:

• We first split the middle term (\(-10fg\)) into two terms \(-f^2g\) and \(-11fg^2\). This is crucial because it allows for grouping. The choice of terms here is guided by breaking the middle term into parts whose product equals the product of the first and last terms of the original expression.
• Grouping these terms, we had two pairs: \([f^2 - f^2g]\) and \([11fg^2 - 11g^2]\).
• The next step was to factor out the common factor from each pair, resulting in \(f^2(1-g) + 11g^2(f-g)\).
• Finally, we factor out the overall common factor, \((1-g)\), from the entire expression, leading to the complete factorization \((1-g)(f^2 + 11g^2)\).

Understanding this method requires recognizing patterns and manipulating polynomial terms to reveal common factors.

Quadratic Expressions in Algebra

Quadratic expressions are a specific type of polynomial where the highest exponent of a variable is two. They are vital in algebra and occur frequently in various types of problems ranging from physics to finance.

The general form of a quadratic expression is \(ax^2 + bx + c\), where \(a, b,\) and \(c\) are constants. The expression \(f^2 - 10fg - 11g^2\) challenges us by arranging two variables, \(f\) and \(g\), within its terms, but remains quadratic due to its highest degree being 2 in terms of single variables.

In dealing with quadratic expressions, several methods can be used for solving or altering them, such as:

• Factoring, which can simplify these expressions by expressing them as a product of simpler terms.
• The Quadratic Formula, useful for finding roots when expressions equate to zero.
• Completing the Square, another method of rearranging the expression to find solutions.

Understanding these methods enables a deeper manipulation and simplification of quadratic expressions in various algebraic contexts.

Checking Factorization

Checking your work is a foundational aspect of mathematics. Factorization can sometimes be tricky, and checking ensures accuracy. Once we've factored an expression, it's crucial to verify that the factorization is correct.

To check that \((1-g)(f^2+11g^2)\) factors the original expression \(f^2 - 10fg - 11g^2\):

• Perform the multiplication of the factors \((1-g)\) and \((f^2+11g^2)\) to expand them back out.
• Calculate: \((1-g)\cdot f^2\) and \((1-g)\cdot 11g^2\), which gives \(f^2 - f^2g + 11g^2 - 11g^3\).
• Rearrange the terms if necessary and compare with the original polynomial. The terms should exactly match \(f^2 - 10fg - 11g^2\).

If the expanded and rearranged expression matches the original, the factorization is confirmed correct. This approach emphasizes understanding and verification, skills that are essential for advanced mathematical work.