Problem 53 - Elementary Algebra

Question

For the following problems, perform the divisions. $$ \frac{y^{3}+5 y^{2}-3}{y-1} $$

Step-by-Step Solution

Answer

Answer: The quotient is $$y^2 + 6y + 6$$, and the remainder is $$3$$.

Step 1: Setup the long division

Write the dividend, $$y^3 + 5y^2 - 3$$, inside the long division bracket and the divisor, $$y-1$$, outside of the bracket.

Step 2: Divide the first term of the dividend by the first term of the divisor

Divide $$y^3$$ by $$y$$. The result is $$y^2$$, which will be the first term of our quotient.

Step 3: Multiply the divisor by the result from Step 2

Multiply the divisor, $$y - 1$$, by $$y^2$$, which results in $$y^3 - y^2$$.

Step 4: Subtract the result from Step 3 from the dividend

Subtract $$y^3 - y^2$$ from $$y^3 + 5y^2 - 3$$. The result is $$6y^2 - 3$$.

Step 5: Repeat the division process with the new expression

Divide the first term of the new expression, $$6y^2$$, by the first term of the divisor, $$y$$. The result is $$6y$$, which is the next term of our quotient.

Step 6: Multiply the divisor by the result from Step 5

Multiply the divisor, $$y - 1$$, by $$6y$$. The result is $$6y^2 - 6y$$.

Step 7: Subtract the result from Step 6 from the new expression

Subtract $$6y^2 - 6y$$ from $$6y^2 - 3$$. The result is $$6y - 3$$.

Step 8: Repeat the division process with the latest expression

Divide the first term of the new expression, $$6y$$, by the first term of the divisor, $$y$$. The result is $$6$$, which is the next term of our quotient.

Step 9: Multiply the divisor by the result from Step 8

Multiply the divisor, $$y - 1$$, by $$6$$. The result is $$6y - 6$$.

Step 10: Subtract the result from Step 9 from the latest expression

Subtract $$6y - 6$$ from $$6y - 3$$. The result is $$3$$, which is the remainder.

Step 11: Write the final quotient and remainder

Our quotient from the long division is $$y^2 + 6y + 6$$, and our remainder is $$3$$. We can write the final answer as: $$ \frac{y^3 + 5y^2 - 3}{y - 1} = y^2 + 6y + 6 + \frac{3}{y - 1} $$

Key Concepts

Understanding Long Division in Polynomials

The process of long division you might remember from arithmetic works similarly with polynomials. When dividing polynomials, we aim to simplify a complex expression into a more manageable form. Here's how it unfolds, step by step:

• Set Up the Problem: In our exercise, we position the polynomial \(y^3 + 5y^2 - 3\) under the division bracket. This is known as the dividend. Outside the bracket, the polynomial \(y - 1\) is our divisor.
• Divide the First Terms: We start by dividing the leading term of the dividend, \(y^3\), by the leading term of the divisor, \(y\). This gives us \(y^2\), which is the first term in our quotient.
• Subtract and Repeat: After multiplying the divisor by \(y^2\), we subtract this from the dividend to get a new expression, then repeat the division process with this new expression until we reach a term we cannot divide the same way.

This repetitive cycle of division, multiplication, and subtraction continues until you achieve a polynomial of lower degree than the divisor, known as the remainder. This step-by-step method ensures you simplify your polynomial systematically.

Interpreting the Remainder

In polynomial division, once the degree of the remaining polynomial is less than the degree of the divisor, we have what's known as the remainder. In our exercise, after a series of divisions and subtractions, we obtain a remainder of \(3\).
The remainder is crucial because it tells us about the closeness of the division. The smaller the remainder, the closer the division result is to being exact. In arithmetic terms, it's akin to the leftover in an integer division, that is, the amount that cannot be divided evenly.
Here's how the remainder fits into the final expression:

• The expression \(\frac{y^3 + 5y^2 - 3}{y - 1}\) can be rewritten with the remainder as \(y^2 + 6y + 6 + \frac{3}{y - 1}\).
• The quotient \(y^2 + 6y + 6\) represents the main part of the division, while \(\frac{3}{y - 1}\) accounts for the remainder.

Understanding the role of the remainder helps you gauge the accuracy and completeness of your division.

Quotient: The Result of Division

The quotient in polynomial division represents the main result of the division process. As we simplified \(y^3 + 5y^2 - 3\) using the divisor \(y - 1\), each step contributed to building our quotient, \(y^2 + 6y + 6\).

• Building the Quotient: We achieved the quotient by continually dividing the leading terms of the polynomial and updating our expression step by step.
• Final Representation: The final represented form of the division is much simpler and easier to handle algebraically, expressed as \(y^2 + 6y + 6\).

In long division, the quotient answers the question, "How many times does the divisor go into the dividend?" It gives us a simplified view of what happens when the entire polynomial is divided efficiently by another, providing key insights into the nature of the original polynomial and its relationship with the divisor.