Problem 110 - Introductory Algebra for College Students

Question

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$3 x^{4}-9 x^{3} y+3 x^{2} y^{2}$$

Step-by-Step Solution

Answer

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

Step 1: Factor out common terms

Looking at the polynomial given, we see that 3 is a common factor in all three terms, meaning we can factor it out. Also, each term contains \( x^{2} \). So we factor out \( 3x^{2} \) from each term, giving us \( 3x^{2}(x^{2} - 3xy + y^{2}) \)

Step 2: Factor the quadratic

The expression inside the parentheses is a quadratic in x and y. Using the formula \( (a - b)^{2} = a^{2} - 2ab + b^{2} \), we can see that the quadratic can be factored into \( (x - y)^{2} \). Thus, the factored form of the original expression is \( 3x^{2}(x - y)^{2} \)

Step 3: Multiplication check

To ensure our factoring works, we can expand the factored form and reduce, confirming that we obtain our original polynomial. Doing this, we see that indeed \( 3x^{2}(x - y)^{2} \) simplifies to \( 3 x^{4}-9 x^{3} y+3 x^{2} y^{2} \), so our factoring is accurate.

Key Concepts

Factoring Polynomials

Factoring polynomials is like breaking down a complex problem into simpler pieces. Imagine trying to find the prime factors of a number. Similarly, with polynomials, we look for ways to simplify or "factor" them into smaller expressions that, when multiplied together, give the original polynomial.

In our example, the polynomial is \(3x^4 - 9x^3y + 3x^2y^2\). The first step in factoring polynomials is to identify common factors in each term. Here, the number 3 is a common factor, and so is \(x^2\). By pulling out these common factors, we reduce the polynomial to \(3x^2(x^2 - 3xy + y^2)\).

This process not only simplifies the polynomial, making it easier to handle, but it also reveals any special structures like difference of squares or perfect square trinomials. Factoring out common terms is usually the fundamental first step in polynomial simplification.

Quadratic Expressions

A quadratic expression is typically in the form \(ax^2 + bx + c\). It involves a degree of two, meaning the highest power of x is squared. Quadratic expressions can appear daunting at first, but once you spot the pattern, they become much more manageable.

For the given problem, after we factor out \(3x^2\), we're left with \(x^2 - 3xy + y^2\). This expression seems complicated, but notice it fits the pattern of a perfect square trinomial, anything in the form \(a^2 - 2ab + b^2\), which is a common pattern in quadratics.

Recognizing this, we factor the expression into \((x-y)^2\). This indicates that our quadratic can be expressed as the square of a binomial. Understanding how to recognize these patterns is pivotal in working with quadratic expressions efficiently.

Multiplying Polynomials

Now that the expression is factored as \(3x^2(x-y)^2\), we perform a multiplication check to ensure the accuracy of the factorization. Multiplying polynomials involves distributing each term across the others, making sure nothing is left out.

By expanding \(3x^2(x-y)^2\), start by squaring \((x-y)^2\) which gives \(x^2 - 2xy + y^2\). Then, we multiply this result by \(3x^2\):

• Multiply \(3x^2\) by \(x^2\) to get \(3x^4\).
• Multiply \(3x^2\) by \(-2xy\) to get \(-6x^3y\).
• Multiply \(3x^2\) by \(y^2\) to get \(3x^2y^2\).

Adding these terms together, we see that the expression matches the original given polynomial: \(3x^4 - 9x^3y + 3x^2y^2\). This check is crucial as it confirms our factoring method was performed correctly, and that each step is accurate.