Problem 36 - Algebra and Trigonometry

Question

\(35-82\) Factor the expression completely. $$ 30 x^{3}+15 x^{4} $$

Step-by-Step Solution

Answer

The completely factored expression is \(15x^3(2 + x)\).

Step 1: Identify Common Factors

Start by identifying any common factors in each term of the expression. Notice that both terms, \(30x^3\) and \(15x^4\), include the factor of \(15x^3\).

Step 2: Factor Out the Common Factor

Factor \(15x^3\) out of the expression. This means you divide each term by \(15x^3\): \[30x^3 + 15x^4 = 15x^3(2 + x)\]

Step 3: Verify Complete Factorization

Check if the terms inside the parentheses, \(2 + x\), can be further factored. Since there are no like terms or further factors, the expression is completely factored.

Key Concepts

Understanding the Common Factor

When factoring polynomials, the first step is to look for a common factor among the terms. A common factor is a number or algebraic term that divides into each term of the polynomial. Identifying this helps to simplify the polynomial. For the expression given, \(30x^3 + 15x^4\), the common factor is \(15x^3\). This is because both terms are divisible by 15 and include powers of \(x\) that can be simplified. Recognizing common factors is like finding a common denominator in a fraction; it simplifies the problem and makes solving much easier. Always start by seeing if there is something each term in the polynomial can "share".

Exploring Polynomial Division

Polynomial division is the process of dividing one polynomial by another. It comes into play after identifying the common factor. Once you find the common factor, you divide each term of the polynomial by this factor, as with simple arithmetic division. For our expression, dividing \(30x^3\) and \(15x^4\) by \(15x^3\) gives us the simplified expression \(15x^3(2 + x)\). This step allows us to pull out the common factor and leaves a simpler expression inside the parentheses. This process is crucial in breaking down complicated expressions into more manageable parts. By simplifying polynomials through division, the underlying structure becomes clear, facilitating further steps.

Achieving Complete Factorization

Complete factorization means breaking down a polynomial into its simplest form where no further factoring is possible. After factoring out the common factor from \(30x^3 + 15x^4\), we obtain \(15x^3(2 + x)\). The task now is to check whether \(2 + x\) can be factored further. In this instance, \(2 + x\) cannot be simplified or factored any further since there are no common factors left within the expression. This means the factorization is complete. A completely factored polynomial is one that's expressed entirely in terms of its factors. It's akin to a prime factorization in numbers, where the polynomial can't be simplified further into other polynomials.