Problem 46 - College Algebra with Corequisite Support

Question

For the following exercises, factor the polynomials. \(3 t(10 t+3)^{\frac{1}{3}}+7(10 t+3)^{\frac{4}{3}}\)

Step-by-Step Solution

Answer

The polynomial factors into \((10t+3)^{\frac{1}{3}}(73t + 21)\).

Step 1: Identify the Common Factor

Firstly, examine both terms \(3t(10t+3)^{\frac{1}{3}}\) and \(7(10t+3)^{\frac{4}{3}}\) for a common factor. Notice that \((10t+3)^{\frac{1}{3}}\) is present in both terms, making it a common factor. The smallest power of this common factor is \((10t+3)^{\frac{1}{3}}\).

Step 2: Factor Out the Common Factor

Factor \((10t+3)^{\frac{1}{3}}\) out of each term: \[ 3t(10t+3)^{\frac{1}{3}} + 7(10t+3)^{\frac{4}{3}} = (10t+3)^{\frac{1}{3}} \left( 3t + 7(10t+3) \right) \]

Step 3: Simplify the Expression

Simplify the expression inside the parentheses: \[ 3t + 7(10t+3) = 3t + 70t + 21 \] Combine like terms: \[ 73t + 21 \]

Step 4: Write the Final Factored Expression

Combine the common factor and the simplified expression to write the fully factored form of the polynomial: \[ (10t+3)^{\frac{1}{3}}(73t + 21) \]

Key Concepts

Understanding the Common Factor

When dealing with polynomials, identifying a common factor is a key step in the factoring process. A common factor is an expression that is present in multiple terms of a given polynomial. This factor can then be factored out to simplify the polynomial.
In our exercise, both terms, \(3t(10t+3)^{\frac{1}{3}}\) and \(7(10t+3)^{\frac{4}{3}}\), share the expression \((10t+3)^{\frac{1}{3}}\) as a common factor. The exponent of the common factor is critical because it's the smallest power that appears in all the terms. This helps in maintaining a common ground when factoring it out.
Once identified, this common factor aids in simplifying the expression, as it provides a means to unify the different parts of the polynomial. By focusing on such shared elements, one can more easily manipulate and simplify complex algebraic expressions.

What Are Algebraic Expressions?

Algebraic expressions are combinations of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. They are like sentences in the language of mathematics and are used to represent relationships and solve problems.
In this particular exercise, the algebraic expression \(3t(10t+3)^{\frac{1}{3}} + 7(10t+3)^{\frac{4}{3}}\) is composed of two terms. Each term is a product involving constants, the variable \(t\), and an expression raised to a power.

• The expression \(3t(10t+3)^{\frac{1}{3}}\) involves a constant (3), a variable \(t\), and \((10t+3)^{\frac{1/3}}\).
• The other term, \(7(10t+3)^{\frac{4/3}}\), consists of the constant 7 and the expression \((10t+3)\) raised to the power of \(\frac{4}{3}\).

Understanding these components is essential for recognizing patterns and performing operations such as factoring. This knowledge also establishes a foundation for more advanced concepts in algebra.

Simplifying Expressions: Bringing Clarity

Simplifying expressions involves reducing them to their simplest form without changing their value or meaning. This process is crucial in algebra as it makes expressions easier to work with and understand.
In the exercise, after factoring out the common factor \((10t+3)^{\frac{1}{3}}\), the remaining expression inside the parentheses, \(3t + 7(10t+3)\), needs simplifying. This step involves distributing the term 7 into \(10t + 3\) to get \(70t + 21\).
Next, combine like terms—those terms that share the same variable part. Here, this means adding \(3t\) and \(70t\) together to get \(73t\). The expression inside the parentheses finally becomes \(73t + 21\).

• Simplification removes redundancies and clarifies the structure of an expression.
• It highlights important relationships or characteristics that may not be obvious initially.

This process transforms complex-looking expressions into a more manageable form that can be easily interpreted and further analyzed or resolved.