Problem 46 - College Algebra

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 as \((10t+3)^{\frac{1}{3}}(73t + 21)\).

Step 1: Identify Common Factors

The expression given is \( 3t(10t+3)^{\frac{1}{3}} + 7(10t+3)^{\frac{4}{3}} \). Both terms have a common factor of \((10t+3)^{\frac{1}{3}}\). We will factor this out in the next step.

Step 2: Factor out the Greatest Common Factor

Factor out \((10t+3)^{\frac{1}{3}}\) from both terms in the expression:\[(10t+3)^{\frac{1}{3}} \left[ 3t + 7(10t+3) \right]\]

Step 3: Simplify Inside the Bracket

Expand and simplify the expression inside the brackets:\[ 7(10t+3) = 70t + 21 \]Substitute back into the factored expression:\[(10t+3)^{\frac{1}{3}} \left[ 3t + 70t + 21 \right]\]

Step 4: Combine Like Terms

Combine the terms inside the bracket:\[ 3t + 70t = 73t \]So the expression becomes:\[(10t+3)^{\frac{1}{3}} (73t + 21)\]

Step 5: Verify the Factorization

Check if the factorization is correct by expanding back to the original expression:Starting with:\[(10t+3)^{\frac{1}{3}}(73t + 21)\]Expand:\[73t (10t+3)^{\frac{1}{3}} + 21 (10t+3)^{\frac{1}{3}} = 3t (10t+3)^{\frac{1}{3}} + 7(10t+3)^{\frac{4}{3}}\]The factorization is correct.

Key Concepts

Greatest Common Factor

In polynomial expressions, the Greatest Common Factor, often abbreviated as GCF, is the largest factor that divides two or more terms. The GCF plays a significant role when factoring polynomials, as it can simplify complex expressions by reducing them to simpler forms.
To find the GCF of a polynomial, evaluate all terms and identify shared factors. For the expression \(3t(10t+3)^{\frac{1}{3}} + 7(10t+3)^{\frac{4}{3}}\), the term \((10t+3)^{\frac{1}{3}}\) appears within both terms.

• Step 1: Look for common terms shared among the components.
• Step 2: Factor out the common term, the GCF. In our case, it's \((10t+3)^{\frac{1}{3}}\).

Once you factor the GCF out, the expression becomes simpler, easing further simplification and combination processes.

Combine Like Terms

Combining like terms is integral when working with expressions since it involves simplifying expressions by summing coefficients of identical terms. This process makes an expression more manageable and compact. In our specific exercise, after factoring out the GCF \((10t+3)^{\frac{1}{3}}\), the expression inside the bracket is \(3t + 70t + 21\).

• Start by identifying terms with identical variables and exponents; like terms have the same variables raised to the same power.
• Next, sum their coefficients: \(3t + 70t = 73t\).

This refinement results in fewer terms and clarity, eventually leading to a more succinct version of the expression.

Simplifying Expressions

Simplifying expressions is the final step in tidying up an equation or expression. It involves organizing the terms in the most concise and comprehensible manner. Once you've found the GCF and combined like terms, simplifying involves checking for any further simplification opportunities.
In the problem presented, after combining \(3t\) and \(70t\) to \(73t\), the polynomials were reformed inside the bracket as \( (10t+3)^{\frac{1}{3}}(73t + 21) \), revealing a cleaner vision of the expression.

• Ensure that all terms are in their simplest form.
• Double-check by expanding to see if the expression matches the original equation, confirming the transformation's accuracy.

Simplification removes unneeded complexity and ensures the expression is ready for any further mathematical computation or analysis.