Problem 32 - College Algebra with Corequisite Support

Question

For the following exercises, factor the polynomial. \(121 x^{2}-88 x+16\)

Step-by-Step Solution

Answer

The polynomial factors to \((11x - 4)^2\).

Step 1: Identify a Pattern

The given polynomial is a trinomial of the form \(ax^2 + bx + c\), where \(a = 121\), \(b = -88\), and \(c = 16\). Notice that 121 is a perfect square (\(11^2\)) and 16 is also a perfect square (\(4^2\)). This suggests that the polynomial might be a perfect square trinomial of the form \((px + q)^2\).

Step 2: Check for a Perfect Square Trinomial

Assuming the trinomial is a perfect square, the middle term should be twice the product of the square roots of the first and last terms. Calculate \(2 \times 11 \times 4 = 88\). Since the middle term is \(-88x\), which is \(-2 \times 11 \times 4\), the trinomial indeed matches the form of a perfect square trinomial \((px + q)^2 = p^2x^2 + 2pqx + q^2\), so it factors to \((11x - 4)^2\).

Step 3: Verify the Factorization

Expand \((11x - 4)^2\) to ensure correctness. The expansion yields \((11x - 4)(11x - 4) = 11x(11x) + 11x(-4) + (-4)(11x) + (-4)(-4)\). This simplifies to \(121x^2 - 44x - 44x + 16\), which further simplifies to \(121x^2 - 88x + 16\). The original polynomial and the expanded polynomial are the same, verifying our factorization.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of polynomial that can be expressed as the square of a binomial. When dealing with trinomials like the one in the exercise, which is written as \(ax^2 + bx + c\), identifying a perfect square pattern is the key.

Here are some important points about perfect square trinomials:

• The first term \(a\) and the last term \(c\) must be perfect squares. In our exercise, \(121 = 11^2\) and \(16 = 4^2\); both are perfect squares.
• The middle term \(b\) should be twice the product of the square roots of the first and last terms. For example, \(b = -88\) was found to be \(-2 \times 11 \times 4\), matching our perfect square pattern requirement.

Once you identify this pattern, the trinomial can be factored into \((px + q)^2\) effectively. Recognizing perfect square trinomials helps you quickly simplify and work with polynomials.

Factoring Techniques

Factoring is an essential skill in algebra that allows us to break down complex expressions into simpler, more manageable ones.

The specific technique used in this exercise involves identifying the polynomial as a perfect square trinomial. However, other techniques also exist for factoring polynomials:

• Difference of Squares: Used when a polynomial can be expressed as \(a^2 - b^2 = (a + b)(a - b)\).
• Grouping: Often used in polynomials with four terms, where terms are grouped in pairs to factor out common factors.
• General Trinomial Factoring: Involves factoring polynomials in the form \(ax^2 + bx + c\) by finding two numbers that multiply to \(ac\) and add to \(b\).

Choosing the right technique depends on the structure of the polynomial. In the case of perfect square trinomials, leveraging the formula \((px + q)^2 = p^2x^2 + 2pqx + q^2\) is a direct and efficient method.

College Algebra

College Algebra covers a broad range of topics, building on concepts learned in high school and preparing students for more advanced mathematics.

One crucial aspect of college algebra is mastering polynomial manipulation and factorization. This exercise highlights a fundamental skill that students must develop:

• Recognizing special polynomial forms, such as perfect square trinomials, is a skill that simplifies solving polynomial equations.
• In-depth understanding of factoring techniques empowers students to solve equations and inequalities with ease.
• Algebraic proficiency forms the foundation for calculus and other higher-level math subjects, making mastery essential.

Practicing exercises like these enhances a student's problem-solving capabilities and prepares them for academic success in mathematics and other disciplines.