Problem 36 - Elementary Algebra

Question

In the following problems, the first quantity represents the product and the second quantity represents a factor of that product. Find the other factor. $$ 0.0024 x^{4 n} y^{3 n+5} z^{2}, 0.03 x^{3 n} y^{5} $$

Step-by-Step Solution

Answer

Answer: The other factor is $0.08 x^n y^{3n} z^2$.

Step 1: Divide coefficients

First, divide the coefficients: $$ \frac{0.0024}{0.03} = 0.08 $$The coefficient of the other factor is 0.08.

Step 2: Divide the x-variable

Now, let's divide the powers of \(x\): $$ \frac{x^{4n}}{x^{3n}} = x^{(4n-3n)} = x^n $$Thus, the x-variable in the other factor is \(x^n\).

Step 3: Divide the y-variable

Next, we will divide the powers of \(y\): $$ \frac{y^{3n+5}}{y^5} = y^{(3n+5-5)} = y^{3n} $$So, the y-variable in the other factor is \(y^{3n}\).

Step 4: Divide the z-variable

Finally, we will divide the powers of \(z\): $$ \frac{z^2}{1} = z^2 $$The z-variable in the other factor is \(z^2\) since there's no \(z\) in the given factor.

Step 5: Combine the results

Combine the coefficient and variables to produce the other factor: $$ 0.08 x^n y^{3n} z^2 $$Hence, the other factor is \(0.08 x^n y^{3n} z^2\).

Key Concepts

Division of Polynomials

When dividing polynomials, think of it like breaking down a complex expression into simpler parts. A polynomial is an algebraic expression with multiple terms. Each term consists of numbers and variables raised to certain powers.
Polynomial division is similar to numerical division. You are essentially finding out how many times one polynomial (the divisor) fits into another (the dividend). The result is called the quotient.

• Step 1: Divide the coefficients. These are the numbers in front of the variables. For example, if dividing 0.0024 by 0.03, you would simply divide these numbers as you would normal numbers, resulting in 0.08.
• Step 2: Divide each variable's exponents separately. For example, when dividing the variable parts like \(x^{4n}\) by \(x^{3n}\), subtract the exponents \(4n - 3n\) to get \(x^n\).
• Step 3: Combine the results to find the quotient, which in this scenario is the 'other factor'.

By mastering each of these steps, you can solve division of polynomial problems with ease.

Exponents

Exponents are a way of representing repeated multiplication. When you see something like \(x^n\), it means that the variable \(x\) is multiplied by itself \(n\) times.
Exponents follow certain basic rules, especially when you are dividing terms with the same base:

• Rule of Subtraction: When you divide like bases, you subtract the exponents. For example, \(x^a / x^b = x^{a-b}\). Similarly, \(y^{3n+5} / y^5 = y^{(3n+5) - 5} = y^{3n}\).
• Special Cases: When an exponent is zero, the result is always 1, as \(x^0 = 1\), provided \(x\) is not zero.

Understanding these basics can simplify any work with exponents, especially in polynomial division, where you often encounter them.

Coefficients

Coefficients are the numerical parts of the terms of a polynomial. Each term in a polynomial expression consists of a coefficient and a variable part (like \(x^{n}\)). For instance, in \(3x^2\), 3 is the coefficient.
The role of coefficients is crucial in polynomial operations, such as division. Here's why coefficients matter:

• Scaling Effect: During division, coefficients scale down the values of terms. For example, dividing 0.0024 by 0.03 gives a scaled-down coefficient of 0.08, which affects the overall size of the quotient "factor" in polynomials.
• Consistency: Multiplying or dividing coefficients ensures the resulting expression consistently represents a scaled relation among terms.

By breaking down both coefficients and variables, we can understand and perform polynomial division more efficiently.