Problem 72 - Elementary and Intermediate Algebra

Question

Factor each polynomial by factoring out the opposite of the GCF. $$ -30 x^{4} y^{3}+24 x^{3} y^{2}-60 x^{2} y $$

Step-by-Step Solution

Answer

The polynomial factors to \(-6x^2 y(5x^2 y^2 - 4xy + 10)\).

Step 1: Identify the GCF

First, identify the greatest common factor (GCF) of the polynomial terms. Look at the coefficients (-30, 24, -60) and the variables (\(x^2\) is the lowest common power of \(x\), \(y\) is the lowest common power of \(y\)).The GCF of the numbers (-30, 24, -60) is 6, and the variables have \(x^2 y\) as a common factor. Therefore, the GCF is \(6x^2 y\).

Step 2: Change the Sign of the GCF

Since we need to factor out the opposite of the GCF, we should consider \(-6x^2 y\) instead of \(6x^2 y\).

Step 3: Apply the Opposite GCF to the Polynomial

Write the original polynomial as a product of \(-6x^2 y\) and a new polynomial:\[-6x^2 y (k)\]Where \(k\) is the quotient of each original term divided by \(-6x^2 y\).

Step 4: Calculate the Quotient for Each Term

Now, determine \(k\) by dividing each term by \(-6x^2 y\):- Term 1: \(-30x^4y^3\) \[\frac{-30x^4y^3}{-6x^2y} = 5x^2y^2\]- Term 2: \(24x^3y^2\) \[\frac{24x^3y^2}{-6x^2y} = -4xy\]- Term 3: \(-60x^2y\) \[\frac{-60x^2y}{-6x^2y} = 10\]

Step 5: Write the Factored Form

Combine these quotients to express the polynomial in factored form:\[-30x^4y^3 + 24x^3y^2 - 60x^2y = -6x^2y(5x^2y^2 - 4xy + 10)\]

Key Concepts

greatest common factor (GCF)

The greatest common factor, often abbreviated as GCF, is the largest factor that divides two or more numbers or expressions without leaving a remainder. It's like finding a common thread that can be pulled from all the parts of a polynomial.
To find the GCF of a polynomial like \[-30 x^{4} y^{3}+24 x^{3} y^{2}-60 x^{2} y\], we begin by identifying it in both numerical coefficients and variables:

• First, look at the coefficients: the numbers in front of the variables (-30, 24, -60). Find the largest number that divides all these numbers. In this case, the GCF of the coefficients is 6.
• Next, for the variables, pick the lowest powers of each that appear in all terms. Here we have \(x^2\) for \(x\) and \(y\) for \(y\), giving us \(x^2y\).

So, the overall GCF of the polynomial's terms is \(6x^2y\). Easy, right? Remember, looking at both the numbers and variables helps ensure no part of the polynomial is left unfactored. In this practice, though, we're interested in the opposite of the GCF, which we explore further.

polynomial division

Polynomial division is the process of dividing one polynomial by another polynomial. In our case, we're dividing each term of the original polynomial by the opposite of the GCF.
Why the opposite? Because the task requires us to factor out the negative of the GCF, which introduces an extra step to our usual GCF factoring.
So, assume the computed GCF is \(6x^2 y\), we need to use \(-6x^2 y\) as our divisor. This changes the division slightly, flipping the signs of outcomes:

• For \(-30x^4y^3\), compute: \[ \frac{-30x^4y^3}{-6x^2y} = 5x^2y^2 \]
• For \(24x^3y^2\), compute: \[ \frac{24x^3y^2}{-6x^2y} = -4xy \]
• For \(-60x^2y\), compute: \[ \frac{-60x^2y}{-6x^2y} = 10 \]

Each term of the quotient, calculated from division, becomes part of a new polynomial factored from the original.
Getting comfortable with polynomial division involves practice. Just keep aligning terms and simplifying cautiously!

variable exponents

Variable exponents add a layer of complexity to polynomials. In the expression \(-30 x^{4} y^{3}\), \(x\) and \(y\) have exponents, meaning they are raised to certain powers. Exponents show how many times a variable is multiplied by itself.
To manage these in factoring, environments like our exercise use the lowest common powers appearing across all terms to help determine the GCF for variables. This means:

• The power (or exponent) of \(x\) across terms is evaluated: given \(x^4\), \(x^3\), and \(x^2\), the smallest is \(x^2\). Thus, \(x^2\) is used in the GCF.
• Similarly, we look at the exponents of \(y\), which are \(y^3\), \(y^2\), and \(y\). The smallest is \(y\), chosen for the GCF.

Understanding variable exponents is crucial for polynomial operations, as they dictate the behavior and outcomes of operations like multiplication and factoring. Keeping track of powers ensures accurate factorization and problem-solving.