Problem 47 - College Algebra

Question

In Exercises \(41-48,\) factor any perfect square trinomials, or state that the polynomial is prime. $$9 x^{2}-6 x+1$$

Step-by-Step Solution

Answer

The factored form of the trinomial \(9x^2 - 6x + 1\) is \((3x - 1)^2\).

Step 1: Identification

Identify the coefficients for the terms in the quadratic trinomial. Here we have \(9x^2\) (with coefficient 9), \(-6x\) (with coefficient -6), and \(1\) (with coefficient 1).

Step 2: Check for Perfect Square Trinomial

Consider whether the quadratic trinomial fits the pattern of a perfect square trinomial. A perfect square trinomial is in the form \(a^2 + 2ab + b^2\), which can be factored into \((a+b)^2\). In comparing our trinomial, we match our coefficients with the formula and observe that \(a^2 = 9x^2\), therefore \(a = 3x\); \(2ab = -6x\), if \(a = 3x\), the value of \(b\) must be -1 to satisfy the equation as \(2*(3x)*(-1) = -6x\), and \(b^2 = 1\) (which supports \(b = -1\)).

Step 3: Factoring

Having observed that all parts of the perfect trinomial formula match our trinomial, we can successfully factor our trinomial. We replace a and b with the respective values we found (3x and -1). Therefore, the factored form of our trinomial is \((a+b)^2 = (3x-1)^2\).

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a specific type of polynomial that can be recognized by its distinct pattern. It follows the form

• \(a^2 + 2ab + b^2\)

where it can be easily factored into

• \((a+b)^2\).

This means that the trinomial is actually the square of a binomial.
To determine if a quadratic trinomial is a perfect square trinomial, you can start by identifying the squares of the first and last terms:

• Look whether there's a term that is a square, like \(a^2\) and \(b^2\).
• Then, check if the middle term could be expressed as \(2ab\).

If these conditions are met, the trinomial is a perfect square. The ease of factoring perfect square trinomials lies in their predictable form, which allows us to quickly transform them into the square of a single binomial expression.

Quadratic Trinomial

A quadratic trinomial is a specific type of polynomial that features three terms, with the highest exponent being two. It takes the general form:

• \(ax^2 + bx + c\)

where \(a\), \(b\), and \(c\) are coefficients, with \(a eq 0\).
These trinomials are quite common and appear in various algebra problems. Depending on their coefficients, quadratic trinomials can often be factored into the product of two binomials. This process of factoring can unlock valuable information about the roots or solutions of the quadratic equation.
In cases where the trinomial is a perfect square, as we saw in the original exercise, it simplifies to a squared binomial. However, not all quadratic trinomials are perfect squares, and they may sometimes be non-factorable or prime.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into a product of simpler polynomials. This is akin to reversing the distributive property in algebra. The goal is to write a complex polynomial as a product of two or more simpler ones.
In the context of quadratic trinomials, factoring often aims to express them as a product of binomials. It simplifies expressions, making it easier to solve equations, especially quadratic ones.

• The process requires an understanding of different factoring techniques, such as recognizing perfect square trinomials and applying them.
• Additionally, methods like splitting the middle term or using the quadratic formula can aid when straightforward factoring is too complex or isn't possible.

While some polynomials can be factored easily, others may remain irreducible, known as "prime polynomials".

Algebraic Expressions

Algebraic expressions encompass a wide range of mathematical phrases that include numbers, variables, and operators like addition and multiplication. These expressions form the foundation of most algebraic concepts.

• An algebraic expression could be as simple as a variable, such as \(x\), or it could be more complex, involving polynomials like \(9x^2 - 6x + 1\).

These expressions are subject to operations that transform or simplify them, which can include factoring, expanding, and combining like terms.
Understanding the foundation of algebraic expressions paves the way to mastering complex equations and functions. In essence, polynomials are a subset of algebraic expressions and are manipulated in similar ways, allowing us to solve and simplify mathematical problems effectively.