Problem 110 - College Algebra

Question

Factor completely. $$ x^{4}-10 x^{2} y^{2}+9 y^{4} $$

Step-by-Step Solution

Answer

The fully factored form of \(x^{4}-10 x^{2} y^{2}+9 y^{4}\) is \((x - \sqrt{3}y)(x + \sqrt{3}y)^2\).

Step 1: Identify the structure of the trinomial

The given trinomial \(x^{4}-10 x^{2} y^{2}+9 y^{4}\) resembles the structure of a perfect square trinomial which is \(a^2-2ab+b^2\). In particular, \(a = x^2\), \(b = 3y^2\), and \(2ab = 2*x^2*3y^2 = 10x^{2}y^{2}\). So the trinomial can be rewritten as \((x^2)^2 - 2*x^2*3y^2 + (3y^2)^2\).

Step 2: Apply Factoring Rules

Using the perfect square trinomial factoring formula, \((x^2)^2 - 2*x^2*3y^2 + (3y^2)^2\) then transforms to \((x^2 - 3y^2)^2\).

Step 3: Apply Difference of Squares Formula

\((x^2 - 3y^2)^2\), is also a square. We can apply the difference of squares formula here as done so previously to rewrite the above expression as \((x - \sqrt{3}y)(x + \sqrt{3}y)^2\).

Key Concepts

Perfect Square Trinomial

In algebra, a perfect square trinomial is an expression that can be written as the square of a binomial. It typically takes the form of \(a^2 \pm 2ab + b^2\). When factoring perfect square trinomials, it's crucial to recognize this pattern in order to simplify the expression correctly.

In our exercise \(x^{4}-10x^{2}y^{2}+9y^{4}\), we identified it as a perfect square trinomial by matching it to the form \(a^2 - 2ab + b^2\), with \((x^2)\) as our \(a\) and \((3y^2)\) as our \(b\). Therefore, factoring this expression yielded the square of the binomial \((x^2 - 3y^2)\).

Understanding this concept is critical, as recognizing such patterns greatly simplifies the process of factoring.

Difference of Squares Formula

The difference of squares is another powerful tool in algebra. It refers to a pattern where the expression \(a^2 - b^2\) can be factored into \(a - b)(a + b)\). This technique is particularly useful for factoring expressions where two terms are squares and are subtracted from one another.

Applying this to our expression \(x^2 - 3y^2\)^2, which itself is a square, we take the square root of each term and apply the difference of squares formula to factor it further into \(x - \sqrt{3}y)(x + \sqrt{3}y)\). Recognizing expressions that fit the difference of squares pattern enables quick and efficient factoring of certain polynomials.

Algebraic Expressions

An algebraic expression represents a mathematical phrase that can include numbers, variables, and operation signs. Expressions are the building blocks of algebra and are used to describe relationships and changes.

The expression we've been examining, \(x^{4}-10x^{2}y^{2}+9y^{4}\), is a complex algebraic expression made up of variables to the power of 4, implying it is a quartic. Simplifying algebraic expressions requires careful observation and a grasp of algebraic structures, like recognizing the potential for perfect square trinomials within them.

Polynomial Factoring Rules

Understanding polynomial factoring rules is essential for breaking down complex expressions into simpler, more manageable pieces. There are multiple factoring techniques, including factoring out the greatest common factor, factoring by grouping, recognizing special products like perfect square trinomials and difference of squares, and using the factoring method suitable to the expression at hand.

With our given polynomial, we used two rules: first, we factored it as a perfect square trinomial, then we applied the difference of squares rule. Mastering these rules can ease the process of managing higher-level algebra problems, such as those involving quartic expressions like ours.