Problem 31 - College Algebra with Corequisite Support

Question

For the following exercises, factor the polynomial. \(49 n^{2}+168 n+144\)

Step-by-Step Solution

Answer

The polynomial factors to \((7n + 12)^2\).

Step 1: Identify the Form

The given polynomial is a quadratic trinomial: \(49n^2 + 168n + 144\). It can be factored using the method of perfect square trinomials or general trinomial factoring.

Step 2: Look for Perfect Square Pattern

The expression \(49n^2 + 168n + 144\) resembles a perfect square trinomial because it can be written in the form \((an + b)^2 = a^2n^2 + 2abn + b^2\). Note that \(49n^2 = (7n)^2\) and \(144 = 12^2\).

Step 3: Verify Middle Term

Verify that the middle term, \(168n\), fits the pattern of a perfect square trinomial. For the perfect square \((7n + 12)^2\), the middle term should be \(2 \times (7n) \times 12\). Calculate: \[2 \times 7 \times 12 = 168.\] This confirms the middle term is correct.

Step 4: Factor as a Perfect Square

Given that the pattern for \(168n\) holds, factor the polynomial as a perfect square trinomial: \((7n + 12)^2\).

Key Concepts

Quadratic Trinomials

Quadratic trinomials are algebraic expressions that take the form: \(an^2 + bn + c\). They often appear in polynomial equations and consist of three terms. Each term includes:

• The first term, \(an^2\), with a quadratic variable.
• The second term, \(bn\), which is linear.
• The third term, \(c\), a constant term.

To factor these expressions, we separate them into simpler binomial products, which can be especially easy when a specific pattern, like a perfect square trinomial, appears. Recognizing such patterns helps to simplify the factoring process, allowing for more efficient problem-solving.

Perfect Square Trinomials

Perfect square trinomials are special types of quadratic trinomials. They are neatly organized and take the form:

• The square of a binomial, such as \((an + b)^2\).

This expands to give a trinomial \(a^2n^2 + 2abn + b^2\). Recognizing the perfect square involves identifying two key elements:

• The first term is a perfect square, like \((an)^2\).
• The last term is a perfect square, such as \(b^2\).

In this case, it resembles the exercise example, \(49n^2 + 168n + 144\), where both 49 and 144 are perfect squares. Spotting such formulas can lead directly to concise factoring of the expression into a binomial square.

Polynomial Factoring Methods

Factoring polynomials involves breaking them down into simpler expressions that, when multiplied, give the original polynomial. Different methods can be applied depending on the form of the polynomial:

• Factor by Grouping: Useful when the polynomial can be split into groups that have a common factor.
• Factoring Quadratic Trinomials: Often involves splitting the middle term or using the AC method.
• Special Forms: Recognizing perfect square trinomials or difference of squares can simplify factoring to a binomial form.

Choosing the right method depends on recognizing specific patterns and structures in the polynomial, crucial for effectively simplifying or solving equations.

Middle Term Verification

Middle term verification ensures the expression is factored accurately, particularly for perfect square trinomials. For a trinomial \(a^2n^2 + 2abn + b^2\), the middle term should equal \(2abn\). Here's how to verify:

• Calculate \(2ab\) for the supposed perfect square.
• Match it with the actual middle term.
• If they equal, it confirms the trinomial can be rewritten as the square of a binomial: \((an + b)^2\).

In the given exercise, the middle term \(168n\) was verified as \(2 \times 7 \times 12 = 168\), thus confirming the trinomial as a perfect square, \((7n + 12)^2\). Confirming the middle term accuracy is an assurance that the factorization is correct and valid.