Problem 51 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 15 x^{3} y^{5}-25 x^{4} y^{2}+10 x^{6} y^{4} $$

Step-by-Step Solution

Answer

The factored form is \(5x^3y^2(3y^3 - 5x + 2x^3y^2)\).

Step 1: Identify Common Factors

First, look for the greatest common factor (GCF) among all the terms in the polynomial. Each term has a numerical coefficient, powers of \(x\), and powers of \(y\). The GCF of the coefficients 15, 25, and 10 is 5. The lowest power of \(x\) across the terms is \(x^3\), and the lowest power of \(y\) is \(y^2\). Therefore, the GCF of the polynomial is \(5x^3y^2\).

Step 2: Factor Out the GCF

Divide each term of the polynomial by the GCF \(5x^3y^2\). This gives: \(\frac{15x^3y^5}{5x^3y^2} = 3y^3\), \(\frac{-25x^4y^2}{5x^3y^2} = -5x\), and \(\frac{10x^6y^4}{5x^3y^2} = 2x^3y^2\). Thus, the factored form is \(5x^3y^2(3y^3 - 5x + 2x^3y^2)\).

Step 3: Verify Factored Form

To ensure the factoring is correct, distribute the GCF back into the expression: \(5x^3y^2(3y^3) = 15x^3y^5\), \(5x^3y^2(-5x) = -25x^4y^2\), \(5x^3y^2(2x^3y^2) = 10x^6y^4\). The original polynomial is retrieved, confirming the factoring is accurate.

Key Concepts

Greatest Common Factor (GCF)

The Greatest Common Factor, or GCF, is a crucial concept in polynomial factoring. It refers to the largest expression that divides each term of a polynomial without leaving a remainder. In our example, we find the GCF of the polynomial \(15x^3y^5 - 25x^4y^2 + 10x^6y^4\).

• First, identify the GCF of the coefficients: the numbers 15, 25, and 10 have a GCF of 5, as 5 is the largest number that can divide each of these coefficients.
• Next, look at the powers of each variable. The lowest power of \(x\) present in each term is \(x^3\), and for \(y\), it is \(y^2\).

When factoring the polynomial, factor out \(5x^3y^2\), which simplifies the expression greatly by reducing each term in the polynomial.

Algebraic Expressions

Algebraic expressions form the basis of algebra and consist of numbers, variables, and arithmetic operations. The polynomial given is an algebraic expression composed of multiple terms consolidated through addition or subtraction. Understanding its components can make factoring more intuitive.

In the polynomial \(15x^3y^5 - 25x^4y^2 + 10x^6y^4\):

• Each term is a product of a numerical coefficient and variables raised to certain powers, such as \(15x^3y^5\).
• Variables \(x\) and \(y\) are raised to different powers in each term, which determines how we find common factors and how terms relate during the factoring process.

Simplifying these expressions by factoring can reveal insights about how the variables and coefficients interact. This is especially helpful when solving algebraic equations or simplifying expressions in more complex algebraic operations.

Polynomial Verification

Verifying the factored form of a polynomial ensures that the original expression is correctly represented by its factors. Here's how we confirm the factoring of our example:

After factoring the polynomial to \(5x^3y^2(3y^3 - 5x + 2x^3y^2)\), distribute the GCF \(5x^3y^2\) back to each term inside the parentheses to check accuracy:

• \(5x^3y^2 imes 3y^3 = 15x^3y^5\)
• \(5x^3y^2 imes (-5x) = -25x^4y^2\)
• \(5x^3y^2 imes 2x^3y^2 = 10x^6y^4\)

When each product equals the corresponding term in the original polynomial, it verifies that the factorization is correct. Using this method of polynomial verification is a reliable way to confirm that no errors were made during the factoring process.