Problem 70 - Introductory Algebra for College Students

Question

Factor completely. $$25 x^{3}-10 x^{2}+x$$

Step-by-Step Solution

Answer

The factorized form of the expression: \(x*(5x-1)*(5x-1)\)

Step 1: Identify the GCF

First, look at all the terms present in the expression \(25x^{3}-10x^{2}+x\). The GCF in all terms is \(x\).

Step 2: Factor out the GCF

Take out the GCF from the polynomial. Doing so, gives \(x(25x^{2}-10x+1)\).

Step 3: Factorize the remaining expression

We notice that the remaining expression inside the brackets \(25x^{2}-10x+1\) is a quadratic factor that can be factored as \((5x-1)^{2}\). Therefore, the factorized form will be \(x*(5x-1)*(5x-1)\).

Key Concepts

Greatest Common Factor

The greatest common factor (GCF), also known as the greatest common divisor, is the largest factor that divides two or more numbers. In the context of polynomials, it refers to the highest degree of any term that can be factored out from each term in the polynomial. Identifying the GCF is a key step in simplifying algebraic expressions and is the first step in polynomial factorization.

For example, consider the expression from our exercise, \(25x^{3}-10x^{2}+x\). Each term contains the variable \(x\) raised to a power and also has numerical coefficients. By examining each term's factors, we find that the GCF is \(x\), as it is the only factor common to all three terms. Once identified, it can be factored out to simplify the polynomial further.

Quadratic Expressions

Quadratic expressions are polynomial expressions that have the highest exponent of 2. The general form of a quadratic expression is \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is nonzero. These expressions may look simple, but they hold significant importance in graphing parabolic curves and solving quadratic equations through various methods such as factoring, completing the square, or using the quadratic formula.

In our exercise, after factoring out the GCF, we were left with a quadratic expression \(25x^{2}-10x+1\). Factoring quadratic expressions requires finding two binomials that multiply to give the original quadratic, which is a critical step in breaking down the expression into its simplest factors.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into its component factors such that when multiplied, they give back the original polynomial. The process often involves identifying the GCF of the terms, factoring out common binomial or trinomial patterns, and applying special formulas, such as the difference of squares or the perfect square trinomial.

In the given exercise, after factoring out the GCF \(x\), the polynomial \(25x^{3}-10x^{2}+x\) was reduced to the quadratic expression \(25x^{2}-10x+1\). This expression was then further factored into \((5x-1)^{2}\), by identifying that it's a perfect square trinomial, where both terms are perfect squares separated by twice the product of their square roots. Such methods are part of a systematic approach to breaking down complex polynomials into easier, more manageable pieces.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations. They can be as simple as a single term, like \(3x\), or as complex as a lengthy polynomial like the one in our exercise. The role of algebraic expressions in mathematics is pivotal as they form the basis for formulating problems and finding solutions.

When working with algebraic expressions, such as the exercise's polynomial, understanding how to manipulate and factor these expressions is fundamental. It not only helps in solving algebra equations but also eases the simplification process, allowing a clearer view of what the expression represents. Mastery of handling algebraic expressions is essential for progress in higher-level math.