Problem 32 - Algebra and Trigonometry

Question

31–76 ? Factor the expression completely. $$ 30 x^{3}+15 x^{4} $$

Step-by-Step Solution

Answer

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

Step 1: Identify Common Factors

First, look for common factors in the expression. Both terms, \(30x^3\) and \(15x^4\), have coefficients and powers of \(x\). The greatest common factor (GCF) is \(15x^3\).

Step 2: Factor Out the Greatest Common Factor

Factor the GCF, \(15x^3\), out of the expression. This gives \(15x^3(2 + x)\).

Step 3: Check for Further Factoring

Check if \(2 + x\) can be factored further. Since \(2 + x\) is a linear expression with no common factors or special factorization, it cannot be simplified further.

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is a critical concept when working with polynomials. It helps to simplify expressions by identifying the largest factor shared by all the terms. Imagine it as the biggest piece you can "factor out" from an expression. For example, when looking at the expression \(30x^3 + 15x^4\), we analyze both terms to find the GCF.

Both coefficients (30 and 15) share a GCF of 15, as it is the largest number that divides both without a remainder. Meanwhile, the variable part \(x\) is common to both terms. The smallest power of \(x\) present is \(x^3\), which means \(x^3\) is part of the GCF.

• Thus, the GCF of the entire expression is \(15x^3\).
• Using the GCF helps to reduce the complexity of the polynomial expression.
• Removing the GCF simplifies the polynomial, making it easier to work with in further calculations.

Polynomial Expressions

Polynomial expressions consist of a sum of terms, with each term being a product of a coefficient and a variable raised to a power. In our example \(30x^3 + 15x^4\), we see two terms:

• \(30x^3\) which represents a single term with a coefficient of 30 and a variable component of \(x^3\).
• \(15x^4\) which is another term with a coefficient of 15 and a variable component of \(x^4\).

Polynomials can vary in complexity:

• They can be simple, with just two terms like a binomial, or more complex with many terms.
• Each term in a polynomial can have different exponents or coefficients.

Factoring polynomial expressions involves breaking down these terms using common factors. This simplification makes it possible to solve equations or understand the characteristics of the polynomial, such as its roots or intercepts.

Linear Expressions

Linear expressions are the simplest form of polynomials. They represent straight-line equations when graphed, typically in the form \(ax + b\). In our step-by-step solution, we encountered a linear expression, \(2 + x\).

• This expression cannot be factored further because it doesn't share any other common factors aside from its basic components.
• Linear expressions are always first-degree polynomials, meaning the highest power of the variable is one.

Understanding linear expressions is essential because they form the building blocks for more complex polynomial expressions.

When simplifying polynomials, ensuring that any linear part is in its simplest form helps to clarify solutions and verify that no further factoring is possible.