Problem 12 - A Graphical Approach to College Algebra

Question

Factor the greatest common factor from each polynomial. $$5(a+3)^{3}-2(a+3)+(a+3)^{2}$$

Step-by-Step Solution

Answer

Factor out \((a+3)\) to get \((a+3)(5(a+3)^{2} + a + 1)\).

Step 1: Identify the Common Factor

Examine each term in the polynomial: \(5(a+3)^{3}\), \(-2(a+3)\), and \((a+3)^{2}\). Notice that each term contains a common factor of \((a+3)\).

Step 2: Determine the Greatest Common Factor

The lowest power of the common factor \((a+3)\) found in the polynomial is \((a+3)\). Thus, \((a+3)\) itself is the greatest common factor (GCF).

Step 3: Factor Out the GCF

Write the polynomial by factoring out the GCF. This involves dividing each term by \((a+3)\), which gives: \((a+3)[5(a+3)^{2} - 2 + (a+3)]\).

Step 4: Simplify the Expression

Now simplify the expression inside the brackets: \((a+3)[5(a+3)^{2} - 2 + (a+3)] = (a+3)[5(a+3)^{2} + (a+3) - 2]\).

Step 5: Combine Like Terms

Finally, simplify further by combining like terms inside the brackets. Compute: \((a+3)[5(a+3)^{2} + (a+3) - 2] = (a+3)[5(a+3)^{2} + a + 3 - 2]\), resulting in \((a+3)[5(a+3)^{2} + a + 1]\).

Step 6: Conclusion

The expression is now factored with the greatest common factor: \((a+3)(5(a+3)^{2} + a + 1)\).

Key Concepts

Greatest Common Factor

Understanding the greatest common factor (GCF) is crucial in simplifying polynomials. The GCF of algebraic expressions is the largest expression that divides each term of the polynomial without a remainder. Here, it involves finding equivalent parts that appear in all components of a polynomial expression.

To find the GCF, follow these steps:

• List any expressions or terms that appear consistently across each term of the polynomial.
• Identify the smallest exponent of the common terms.
• The expression with these terms and the smallest exponent is the GCF.

By applying this to our example, we recognize that \(a+3\) exists in every term of the polynomial: \(5(a+3)^3\), \(-2(a+3)\), and \( (a+3)^2 \). The term \(a+3\), appearing with the smallest exponent, becomes our GCF.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. These expressions are the building blocks in algebra used to represent various mathematical situations.

Elements of algebraic expressions include:

• Constants: Fixed numeric values within the expression.
• Variables: Symbols representing unknown numbers, often denoted by letters such as \(x\), \(y\), or \(a\).
• Coefficients: The numeric part of terms involving variables.
• Operators: Includes symbols like \(+, -, \times, \div\) that indicate operations.

The polynomial in the exercise is composed of terms like \(5(a+3)^3\) and \(-2(a+3)\), where each term showcases the structure of an algebraic expression with both constant and variable elements. Understanding how to manipulate and simplify these expressions is key to mastering algebra.

Polynomial Simplification

Polynomial simplification enables us to write expressions in their simplest form, making them easier to understand and solve. It involves a systematic process of combining like terms and factoring common factors.

Steps for simplifying polynomials include:

• Identify and factor the GCF: Take out any common factors from all terms, as seen previously with \(a+3\) in our example.
• Reorganize the expression: Factor out the GCF, writing the polynomial in a bracketed format. This helps separated terms to be combined or simplified further.
• Combine like terms: Add or subtract terms with the same variables and powers to reduce the expression to simplest form.

In the step-by-step solution, removing \(a+3\) from each term simplified the problem to \( (a+3)(5(a+3)^2 + a + 1) \), presenting it in a more concise and understandable way. With practice, simplifying polynomials becomes a straightforward method for problem-solving in algebra.