Problem 28 - A Graphical Approach to College Algebra

Question

Factor each trinomial completely. $$9 x^{2}-6 x^{3}+x^{4}$$

Step-by-Step Solution

Answer

The trinomial factors to \(x^2(x - 3)^2\).

Step 1: Identify the Greatest Common Factor (GCF)

The terms in the trinomial are \(9x^2\), \(-6x^3\), and \(x^4\). First, find the GCF of these terms. Each term contains at least \(x^2\), so the GCF is \(x^2\).

Step 2: Factor Out the GCF

Factor \(x^2\) out of each term in the trinomial: \[9x^2 - 6x^3 + x^4 = x^2(9 - 6x + x^2)\].

Step 3: Reorder and Recognize a Quadratic

Inside the parentheses, reorder the terms so they form a standard quadratic expression: \(x^2(9 - 6x + x^2) = x^2(x^2 - 6x + 9)\).

Step 4: Factor the Quadratic

Recognize that \(x^2 - 6x + 9\) is a perfect square trinomial, because \((-3)^2 = 9\) and \(2(-3) = -6\). Thus, \(x^2 - 6x + 9 = (x - 3)^2\).

Step 5: Write the Fully Factored Form

Combine the factors to express the trinomial completely factored: \[x^2(x - 3)^2\].

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) of a set of terms is the largest factor that divides each of the terms without leaving a remainder. It is an essential concept when factoring trinomials as it simplifies the expression and paves the way for further factorization.

In the example given, you have the trinomial terms: \(9x^2\), \(-6x^3\), and \(x^4\). Identifying the GCF begins with looking at both numerical and variable components. For the coefficients of these terms, the numbers \(9\), \(6\), and \(1\) have the greatest common divisor of \(1\). However, looking at the variable part, each term contains at least \(x^2\). Hence, the GCF for the entire trinomial is \(x^2\).

By factoring \(x^2\) out of each term, you simplify the trinomial to \(x^2(9 - 6x + x^2)\), making it easier to work with. Simplification using the GCF is often the first and most crucial step in the factoring process.

Perfect Square Trinomial

Recognizing and factoring a perfect square trinomial can greatly simplify expressions. A trinomial becomes a perfect square if it can be expressed in the form \((a + b)^2\) or \((a - b)^2\).

In the example trinomial, after factoring out the GCF, you have the quadratic expression \(x^2 - 6x + 9\). To see if it is a perfect square, consider if it fits \((x - 3)^2\). The middle term must be twice the product of the square roots of the first and last terms, which is true here: \(2(-3) = -6\), and \((-3)^2 = 9\).

This recognition transforms the quadratic expression into \((x - 3)^2\), simplifying the expression significantly. Identifying perfect square trinomials is a powerful technique for swiftly simplifying polynomial expressions.

Quadratic Expression

A quadratic expression is a polynomial with a degree of 2, typically written as \(ax^2 + bx + c\). Understanding the structure of these expressions is crucial when factoring trinomials.

The expression inside our parentheses \(x^2(9 - 6x + x^2)\) can be reorganized to \(x^2(x^2 - 6x + 9)\), fitting the standard form of a quadratic: where \(a = 1\), \(b = -6\), and \(c = 9\).

Quadratics can often be factored further into binomials or, in some cases, recognized as perfect squares or differences of squares. Identifying these forms allows one to factor them easily or recognize patterns that simplify the factoring process, ultimately making problem-solving much easier.