Problem 45 - College Algebra with Corequisite Support

Question

For the following exercises, factor the polynomials. \(3 c(2 c+3)^{-\frac{1}{4}}-5(2 c+3)^{\frac{3}{4}}\)

Step-by-Step Solution

Answer

The polynomial factors to \(- (2 c + 3)^{-\frac{1}{4}} (7 c + 15)\).

Step 1: Identify the Greatest Common Factor (GCF)

The two terms in the expression are \(3 c(2 c+3)^{-\frac{1}{4}}\) and \(-5(2 c+3)^{\frac{3}{4}}\). The common base in the exponents is \((2 c + 3)\). The smallest exponent among the terms is \(-\frac{1}{4}\), which is part of both terms. Therefore, the GCF is \((2 c + 3)^{-\frac{1}{4}}\).

Step 2: Factor Out the GCF

Factor out \((2 c + 3)^{-\frac{1}{4}}\) from the expression:\[ 3 c(2 c + 3)^{-\frac{1}{4}} - 5(2 c + 3)^{\frac{3}{4}} = (2 c + 3)^{-\frac{1}{4}} \left[ 3 c - 5 (2 c + 3) \right] \]

Step 3: Simplify the Factored Expression

Distribute and simplify the expression inside the brackets:\[ 3 c - 5(2 c + 3) = 3 c - 10 c - 15 = -7 c - 15 \]Substitute back:\[ (2 c + 3)^{-\frac{1}{4}} (-7 c - 15) \]

Step 4: Final Factored Form

Combine your results to get the final factored form, which is:\[ - (2 c + 3)^{-\frac{1}{4}} (7 c + 15) \]

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest factor that two or more terms share. When working with polynomials, identifying the GCF is a vital step in simplifying expressions. In our exercise, we have two terms: \(3 c(2 c+3)^{-\frac{1}{4}}\) and \(-5(2 c+3)^{\frac{3}{4}}\). Both terms share a common part:

• The factor \((2c + 3)\) with exponents \(-\frac{1}{4}\) and \(\frac{3}{4}\).

To determine the GCF, we look for the smallest exponent of this common base. Here, the exponent \(-\frac{1}{4}\) is smaller than \(\frac{3}{4}\), making \((2 c + 3)^{-\frac{1}{4}}\) our GCF.
By factoring out this GCF, we simplify the polynomial, initiating the process of making it more manageable for solving or simplifying further.

Exponents

Exponents are a way to express repeated multiplication of a number by itself. They are a key part of mathematics and crucial in algebra, especially in polynomials.

• Exponents dictate the number of times a base number is multiplied by itself.
• Positive exponents indicate repeated multiplication: \(x^2 = x \times x\).
• Negative exponents indicate division or the reciprocal: \(x^{-1} = \frac{1}{x}\).

In our context, each term in the polynomial had \((2c + 3)\) raised to an exponent. By understanding how exponents work, particularly negative exponents, we efficiently factor out and simplify the given polynomial.
This simplification involves using the smallest exponent, which will be our GCF, and adjusting other terms accordingly.

Polynomial Simplification

Polynomial simplification makes complex algebraic expressions more understandable and easier to work with. It involves combining like terms, factoring, and reducing the expression to its simplest form.

• Start by identifying common factors that allow for factoring out, such as the GCF.
• Use the distributive property to simplify expressions within brackets.

In the original solution, once the GCF \((2c + 3)^{-\frac{1}{4}}\) was factored out, the expression inside the brackets \([3c - 5(2c + 3)]\) was simplified.
The brackets contain a small polynomial that was expanded and simplified to \(-7c - 15\). Re-integrating this simplified part back with the factored out GCF gives us our fully simplified polynomial.
Having the polynomial in its simplified form is invaluable for further operations such as solving equations or graphical representations.