Problem 113 - College Algebra

Question

Factor completely. $$ 2 x^{2}-7 x y^{2}+3 y^{4} $$

Step-by-Step Solution

Answer

The factored form of the given polynomial \(2x^2 - 7xy^2 + 3y^4\) is \((x - y^2)(2x - 3y^2)\).

Step 1: Identify the Binomial Expressions

We need to write the polynomial in two binomial expressions: \(2x^2 - 7xy^2 + 3y^4\) = \((ax + by)\) \((cx - dy)\). Let's start by guessing factors of the first and last term. Possible factors for \(2x^2\) could be \(2x \cdot x\) and for \(3y^4\) could be \(3y^2 \cdot y^2\). Now we have to find the right pair that adds up to \(-7xy^2\) with an arrangement of positive and negative signs. After some trials, it can be seen that if \(a = x\), \(b = -y^2\), \(c = 2x\) and \(d = -3y^2\), then the middle term works out to be \(-7xy^2\).

Step 2: Write the Given Polynomial as a Product of Binomials

Add the obtained values of \(a\), \(b\), \(c\) and \(d\) in the binomial expressions: \(2x^2 - 7xy^2 + 3y^4 = (x - y^2)(2x - 3y^2)\)

Step 3: Verify the Factored Form

Multiply the binomial expressions to double-check if the original polynomial is obtained. Upon checking, it is confirmed that the factored form is correct.

Key Concepts

Binomial Expressions

When you think of binomial expressions, imagine a simple yet important mathematical concept that involves two distinct terms linked together. For example,

• \( (x - y^2) \)
• \( (2x - 3y^2) \)

are binomial expressions. Each of these has two terms, unlike a monomial (one term) or a trinomial (three terms).

These expressions form the building blocks when dealing with algebraic equations. They can often be seen as parts of broader expressions, such as in polynomial equations where multiple binomials are multiplied together. Understanding binomials is key as they are foundational in simplifying more complex polynomial expressions and solving algebraic problems.

Quadratic Polynomial

A quadratic polynomial is an equation where the highest degree of the variable is squared. In our example, the quadratic polynomial is \(2x^2 - 7xy^2 + 3y^4\).

A typical form of a quadratic polynomial is \(ax^2 + bx + c\), where:

• \(a\), \(b\), and \(c\) are coefficients
• \(x\) is the variable

However, our polynomial exhibits another variable \(y\), adding complexity. Here, the "leading term" is \(2x^2\), which strongly influences the polynomial's degree. A quadratic polynomial like this can be expressed in terms of products of binomials simplifying calculations and solutions.

Factoring Technique

Factoring is crucial in simplifying and solving polynomial equations efficiently. This technique involves breaking down a complex expression into simpler, multiplied forms.

To factor the quadratic polynomial \(2x^2 - 7xy^2 + 3y^4\), identifying fitting binomials is essential.

• First, we guess suitable factors of the initial and final terms.
• The potential factors for \(2x^2\) are \(2x\) and \(x\), while \(3y^4\) may factor into \(3y^2\) and \(y^2\).
• By adjusting the signs and arrangement, we form the correct middle term \(-7xy^2\).

The goal is to rewrite the polynomial:

• \((x - y^2)(2x - 3y^2)\)

Multiply back these binomials to verify their correctness. This checks if they recreate the original equation, providing an effective solution and proof of factoring accuracy.