Problem 10 - College Algebra

Question

In Exercises \(1-10\), factor out the greatest common factor. $$x^{2}(2 x+5)+17(2 x+5)$$

Step-by-Step Solution

Answer

The factored expression is \((2x+5)(x^2 + 17)\)

Step 1: Identify the common factor

Looking at the given expression, \(x^{2}(2 x+5)+17(2 x+5)\), the term in parenthesis \((2x + 5)\) is the common factor present in both terms.

Step 2: Factorization

Use the distributive property (also known as the factoring out technique) which states that for all real numbers a, b and c: a*(b+c) = a*b + a*c. In this case, factor out \((2x + 5)\) from both terms of the expression, resulting in: \((2x+5)(x^2 + 17)\)

Step 3: Final Expression

The factored expression is \((2x+5)(x^2 + 17)\) after the common factor \((2x + 5)\) has been factored out from the original expression.

Key Concepts

Greatest Common Factor

Understanding the greatest common factor (GCF) is essential for simplifying algebraic expressions and solving problems efficiently. The GCF is the highest number that divides exactly into two or more numbers. When factoring expressions, identifying the GCF of the terms can significantly simplify the problem.

For instance, if we're given the expression in the exercise, finding the GCF means looking for the highest expression that is common to all terms. Here, the binomial \(2x + 5\) is a common factor in both terms of the expression \(x^{2}(2 x+5)+17(2 x+5)\). Recognizing this, we can apply the distributive property to factor out the GCF and simplify the expression, easing our path towards the solution.

Distributive Property

The distributive property, also known as distribution, is a fundamental principle in algebra that describes how multiplication is distributed over addition or subtraction within parentheses. The property states that given constants or variables \(a\), \(b\), and \(c\), we have \(a(b + c) = ab + ac\).

This property is crucial when we are factoring algebraic expressions like in the provided textbook exercise. It allows us to 'factor out' or 'distribute out' the greatest common factor from each term in an expression. By applying this property, we take a compound expression and break it down into simpler parts, facilitating further manipulation or solution of equations.

Algebraic Expression Factorization

Factorization of algebraic expressions involves breaking down complex expressions into simpler, multiplicative parts or 'factors' that, when multiplied together, give back the original expression. We often look for the greatest common factor to begin the factorization process, as it simplifies the expression significantly.

When factorizing expressions such as \(x^{2}(2 x+5)+17(2 x+5)\), we first seek any common factors across terms. After applying the distributive property to 'pull out' the GCF, we can rewrite the entire expression as a product of two simpler expressions, which can be further evaluated or used to solve equations.

Polynomial Factoring

Polynomial factoring is an extension of the concept of factorization to polynomials, which are expressions involving variables raised to whole number exponents. The process of polynomial factoring may involve finding the GCF as the first step, followed by the use of various techniques, such as grouping, the use of special product formulas, or long division, to further factorize the polynomial into irreducible factors.

The exercise supplied presents us with a polynomial that we can factor by identifying the binomial \(2x + 5\) as the common factor. This process of extracting the GCF and restructuring the polynomial is a foundational skill in algebra which aids in the simplification, evaluation, and solving of more complex mathematical problems.