Problem 62 - Introductory Algebra for College Students

Question

Factor any perfect square trinomials, or state that the polynomial is prime. $$x^{2}-18 x y+81 y^{2}$$

Step-by-Step Solution

Answer

The factored form of the perfect square trinomial \(x^{2}-18 x y+81 y^{2}\) is \((x - 9y)^2\).

Step 1: Identify the squared terms

The first and third terms of the trinomial, \(x^{2}\) and \(81y^{2}\), are perfect squares. The roots of these squares are \(x\) and \(9y\) respectively. Hence \(a = x\) and \(b = 9y\).

Step 2: Check the middle term

The middle term \(18xy\) is equal to \(2ab = 2 \cdot x \cdot 9y\). So it confirms that this trinomial is indeed a perfect square.

Step 3: Factor the trinomial

Write the trinomial as the square of a binomial. It is \((a-b)^2\) where \(a=x\) and \(b=9y\). So the factored form is \((x - 9y)^2\).

Key Concepts

Perfect Square Trinomials

A perfect square trinomial is a special type of polynomial that can be factored into a binomial squared. When we say a trinomial is a perfect square, it means that the trinomial is formed by squaring a binomial. This means that it can be expressed in the form (a ± b)².

To identify a perfect square trinomial, you should check if:

• The first and last terms are square numbers. In the expression you provided, x² and 81y² are perfect squares as they are squares of x and 9y, respectively.
• The middle term should be twice the product of the roots of the squares. For the expression, the middle term is -18xy, which is equal to 2ab = 2 imes x imes (-9y).

If these conditions are satisfied, it confirms that the given polynomial is a perfect square trinomial, making it easier to factor.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operational symbols. They are foundational elements in algebra, representing real-world quantities in a formulaic manner. Think of them as the building blocks of algebra.

For example, in the expression x² - 18xy + 81y², we have:

• Three terms: x², -18xy, and 81y².
• Two variables: x and y, whose values can change or be substituted.
• Coefficients: 1 for x², -18 for xy, and 81 for y².

A solid understanding of algebraic expressions ensures that you can manipulate and simplify expressions as required. This skill is crucial for solving a variety of algebraic problems, including factorization.

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into simpler, non-divisible polynomials called factors. This is similar to expressing numbers as a product of prime numbers in arithmetic.

To factor polynomials, try these steps:

• Identify any common factors in the polynomial terms. For some polynomials, factoring out a greatest common factor (GCF) can simplify the problem.
• Look for recognizable forms like perfect square trinomials or difference of squares. These forms allow you to factor quickly using formulas.
• Use methods like grouping or applying special factorizations for polynomials that don’t easily fit standard forms.

In the given exercise, recognizing it as a perfect square trinomial helped instantly factor it into (x - 9y)². Polynomial factorization is a powerful technique used in simplifying expressions, solving equations, and other advanced algebraic applications.