Problem 126 - Introductory Algebra for College Students

Question

Factor each polynomial. $$4 x^{2 n}+12 x^{n}+9$$

Step-by-Step Solution

Answer

The factor of the polynomial \(4 x^{2 n}+12 x^{n}+9\) is \((2x^n + 3)^2\).

Step 1: Identifying Parameters

This is a trinomial, so it can generally be written in the form \(a^{2n}+2ab+b^2\). Upon inspection, in the given expression \(4 x^{2 n}+12 x^{n}+9\), \(a\) is \(2x^n\), as \(a^{2n}\) is \(4x^{2n}\), \(b\) is \(3\), as \(b^2\) is 9, and \(2ab\) is \(12x^n\) upon substitution of \(a\) and \(b\). Thus, the trinomial is in the form of a perfect square.

Step 2: Applying the Perfect Square Trinomial Formula

The rule for a perfect square trinomial is: \(a^{2n}+2ab+b^2 = (a+b)^{2n}\). Therefore, the perfect square trinomial of \(4 x^{2 n}+12 x^{n}+9\) will be \((2x^n + 3)^2\) upon substituting the identified values.

Key Concepts

Understanding Trinomials

Trinomials are a type of polynomial. They consist of three terms, hence the prefix 'tri'. In general, a trinomial will look like this: \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is a variable. Ready to break it down a bit more?

• The 'coefficients' like \(a, b,\) and \(c\) are numbers that stand in front of variables or powers of variables.
• Each term in a trinomial can be a combination of numbers and one or more variables.

Trinomials are commonly encountered in algebra and understanding how to manipulate them is crucial. They serve as foundational elements for more complex algebraic expressions and equations. It's useful to recognize different forms of trinomials because they can give clues about how to simplify or solve them.

Identifying Perfect Square Trinomials

Perfect square trinomials are a special category of trinomials. Recognizing them can help simplify expressions significantly. When looking at an expression like \(a^{2n} + 2ab + b^2\), you might be dealing with a perfect square. Here's why:

• The first term, \(a^{2n}\), represents a square of some term \(a\).
• The last term \(b^2\) is also a perfect square.
• The middle term is two times the product of \(a\) and \(b\).

If an expression fits this template, you can rewrite it as \((a + b)^2\). This is called the perfect square trinomial rule. Remember this trick as it can save time and effort in calculations.

Mastering Polynomial Factoring Steps

Factoring polynomials involves breaking them down into simpler components, which when multiplied together, give the original polynomial. Here’s the general approach to factor a trinomial:

• Identify the Form: First, determine if the trinomial fits any special form, like a perfect square trinomial.
• Find Parameters: For a perfect square trinomial, find the values of \(a\) and \(b\) that satisfy the structure \(a^{2n} + 2ab + b^2\).
• Apply the Formula: Rewrite the trinomial as \((a + b)^2\) if it fits the perfect square form.

Following these steps helps ensure you understand and correctly apply polynomial factoring methods. With practice, identifying and factoring these expressions will become second nature!