Problem 54 - Introductory Algebra for College Students

Question

Factor any perfect square trinomials, or state that the polynomial is prime. $$9 x^{2}+6 x+1$$

Step-by-Step Solution

Answer

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

Step 1: Identify the form of trinomial

The given trinomial is of the form \(ax^2 + bx + c\), where \(a = 9\), \(b = 6\) and \(c = 1\). The trinomial is perfect square if \(b = 2\sqrt{ac}\).

Step 2: Check if the trinomial is perfect square

Substitute \(a\), \(b\) and \(c\) into the formula \(b = 2\sqrt{ac}\), we have \(6 = 2\sqrt{9 * 1}\). Calculating the right side we have \(6 = 6\), which means the trinomial is a perfect square.

Step 3: Factor the perfect square trinomial

Since the trinomial is a perfect square trinomial, it could be factored into \((\sqrt{a}x + \sqrt{c})^2\). Substituting \(a = 9\) and \(c = 1\), the trinomial could be factored into \((3x + 1)^2\).

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental skill in algebra that involves breaking down a polynomial into simpler components, called factors, that multiply together to produce the original polynomial. When you factor a polynomial, you're essentially reversing the process of multiplication. This can make equations simpler to analyze and solve.

In the problem given, the trinomial \(9x^2 + 6x + 1\) is shown to be a perfect square trinomial, allowing it to be factored into \((3x + 1)^2\). This is because each term in the trinomial is part of a structure that matches a recognizable pattern:

• The first term \((9x^2)\) and the constant term \((1)\) are both perfect squares.
• The middle term \((6x)\) connects them through a relationship defined by the perfect square trinomial form: \(b = 2\sqrt{ac}\).

Identifying polynomials as perfect squares is a useful strategy in factoring because it can significantly simplify the process.

Quadratic Expressions

Quadratic expressions are polynomial expressions of the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(x\) is the variable. These expressions are called "quadratic" because the highest degree of the variable is 2.

Understanding the structure of a quadratic expression is crucial for identifying factoring opportunities. For example, the expression \(9x^2 + 6x + 1\) fits the general form, with specific values for \(a\), \(b\), and \(c\) that help determine how the expression can be rewritten as a product of two binomials or recognized as a perfect square trinomial.

When dealing with quadratics, it helps to:

• Identify if the quadratic can be factored into simpler expressions.
• Check for special patterns, like perfect square trinomials or difference of squares.
• Simplify the factoring process by identifying the relationship between the coefficients.

Mastering quadratic expressions allows for easier manipulation and solving of equations related to them.

Algebraic Techniques

Algebraic techniques provide the tools that help simplify and solve polynomial equations. When applied to expressions like quadratics, these techniques allow us to rewrite them in more workable forms. Techniques might include factoring, completing the square, or using the quadratic formula, depending on the expression's structure.

For instance, recognizing that \(9x^2 + 6x + 1\) forms a perfect square is a key algebraic technique that simplifies the problem. By using the identity \((ax^2 + bx + c) = (\sqrt{a}x + \sqrt{c})^2\) when \(b = 2\sqrt{ac}\), we quickly reach the factored form \((3x + 1)^2\).

Common algebraic techniques to keep in mind include:

• Distributing and combining like terms to identify factored forms.
• Employing specific patterns and identities such as perfect squares, difference of squares, or cubics.
• Transforming quadratics into a different form to make solving easier, like converting to vertex form.

Using these techniques skillfully can turn complicated algebraic problems into solvable ones, making algebra a valuable toolset for students.