Problem 110 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The factorized form of the given expression is \((x - 3y)^2(x + 3y)^2\).

Step 1: Identify the terms

The given expression can be written in the form of \(a^2 - 2ab + b^2\) where \(a = x^2\), \(b = y^2\), and \(2ab = 10x^2y^2\). So the expression becomes: \(a^2 - 2ab + b^2\)

Step 2: Apply the formula for a perfect square trinomial

The formula for a perfect square trinomial is \(a^2 - 2ab + b^2 = (a - b)^2\). If we substitute a = x^2 and b = y^2 into this formula, the expression \(x^4 - 10x^2y^2 + 9y^4\) becomes \((x^2 - y^2)^2\)

Step 3: Factorize the perfect square

The inner difference of squares further factorizes into \((x - 3y)(x + 3y)\). The fully factorized form of the expression is thus: \((x - 3y)^2(x + 3y)^2\)

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of polynomial that can be expressed in the form \(a^2 \pm 2ab + b^2\). This structure makes factoring particularly straightforward. When you recognize a polynomial as a perfect square trinomial, you can simplify it using the formula \((a \pm b)^2\). Let's break it down:

• Identify the squares: Ensure that the first and last terms are perfect squares. In our expression \(x^4 - 10x^2y^2 + 9y^4\), \(x^4\) is \((x^2)^2\) and \(9y^4\) is \((3y^2)^2\).
• Check the middle term: It should be twice the product of the square roots of the first and last terms. Here, \(2ab = 10x^2y^2\), fitting the form of \(-2(x^2)(3y^2)\).

By matching the structure \(a^2 - 2ab + b^2\), we can factor the polynomial as a perfect square. That would give us \((x^2 - 3y^2)^2\), simplifying the multiplication and revealing the inner relationships within the polynomial.

Difference of Squares

The difference of squares is a classic algebraic identity that allows further simplification of polynomial expressions. The formula is \(a^2 - b^2 = (a - b)(a + b)\). It requires two terms, both perfect squares, separated by a subtraction sign:

• The term \((x^2 - y^2)\) from our expression is an example of a difference of squares, where \(a = x\) and \(b = 3y\).
• Applying the difference of squares formula to \((x^2 - y^2)\), we factor it as \((x - 3y)(x + 3y)\).

Recognizing the difference of squares within complex polynomials allows you to break them down into simpler binomials. This method is particularly useful for expressions that were initially represented as perfect square trinomials, such as when they have been factored down to or include terms similar to \((x^2 - y^2)\).

Polynomial Factorization

Polynomial factorization involves breaking down a complex polynomial into simpler, non-divisible factors. This process is akin to finding the numbers that, when multiplied together, reproduce the original number, but applied to expressions:

• It helps in simplifying expressions, solving equations, and understanding polynomial identities.
• Given an example like \(x^4 - 10x^2y^2 + 9y^4\), factorization requires several steps. First, identify if it's a special pattern like the perfect square trinomial or the difference of squares.
• After recognizing the special pattern, apply relevant identities to break it down further. Here, we express it as \((x^2 - 3y^2)^2\), and then utilize the difference of squares to factor each component into \((x - 3y)^2(x + 3y)^2\).

Through polynomial factorization, not only do we simplify the expression but gain insight into its underlying structure. This makes solving polynomial equations more manageable and allows polynomial division or finding roots to be more straightforward.