Problem 109 - Introductory Algebra for College Students

Question

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$2 x^{4}+6 x^{3} y+2 x^{2} y^{2}$$

Step-by-Step Solution

Answer

The completely factored form of the given polynomial \(2 x^{4}+6 x^{3} y+2 x^{2} y^{2}\) is \(2x^2(x^2+3xy+y^2)\)

Step 1: Identify Common Factor

Looking at the given polynomial expression \(2 x^{4}+6 x^{3} y+2 x^{2} y^{2}\), it can be observed that each term has a common factor of \(2x^2\).

Step 2: Factor out the Common Factor

We can factor out this common fraction from each term by division: \(2 x^{4}/2x^2 = x^2, 6 x^{3} y/2x^2 = 3xy, 2 x^{2} y^{2}/2x^2 = y^{2}\). Therefore, the factored form of this polynomial is \(2x^2(x^2+3xy+y^2)\)

Step 3: Verification Through Multiplication

Multiplying out the factored form will confirm if our factorization is correct: \(2x^2 * x^2 = 2x^4, 2x^2 * 3xy = 6x^3y, 2x^2 * y^2 = 2x^2y^2\). Hence the multiplication works out correctly, proving our factorization is correct.

Key Concepts

Factoring

Factoring is an essential skill for solving polynomial expressions. It simplifies equations, making them easier to work with. Factoring is like taking a group of terms and finding a common divisor that can be "pulled out" or extracted. When factoring polynomials, the goal is to express the polynomial as a product of simpler polynomials. In our example, the polynomial is:\[2x^4 + 6x^3y + 2x^2y^2\]The first step in factoring is to look for a common factor in all terms of the polynomial. Here, each term includes \(2x^2\). By factoring out \(2x^2\) from each term, the expression simplifies to:\[2x^2(x^2 + 3xy + y^2)\]Now, the original polynomial is presented as a factored expression, which is easier to handle in algebraic expressions and can reveal solutions that might not be obvious at first glance.

Polynomial Expressions

Polynomial expressions are mathematical phrases involving numbers and variables that are combined using addition, subtraction, multiplication, and positive integer exponents. Each "term" in a polynomial is formed by multiplying constants, known as "coefficients," with variables raised to power.In the exercise, the polynomial expression is:

• The terms: \(2x^4\), \(6x^3y\), and \(2x^2y^2\)
• Variables: \(x\) and \(y\)
• Degrees: Highest power is \(x^4\)

When working with polynomial expressions:

• Understand structure: Identify individual terms and their coefficients.


• Identify patterns: Recognize similar structures across terms, such as common factors.


• Simplify when possible: Factoring is one method of simplification that can help solve more complex algebraic equations.

Understanding these characteristics enables students to manipulate and solve equations more effectively.

Algebraic Verification

Algebraic verification is a useful practice to ensure that manipulations and solutions to problems are correct. When you factor a polynomial, you can use multiplication to verify accuracy.For verification:

• Multiply back the factors to see if you reconstruct the original polynomial.


• For our example, multiply the factors \(2x^2(x^2 + 3xy + y^2)\) to check its correctness.
• Carry out term-by-term verification:
  • \(2x^2 \times x^2 = 2x^4\)
  • \(2x^2 \times 3xy = 6x^3y\)
  • \(2x^2 \times y^2 = 2x^2y^2\)

Successful multiplication means the factorization is accurate. This also provides a self-check mechanism to catch errors and solidify your understanding of polynomial expressions and their manipulations.