Problem 5 For the following problems, perf... [FREE SOLUTION]

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Elementary Algebra

Wade Ellis, Jr.

$Math Studyset 91影视 Explanations$ Math

2010 Edition

Chapter 8: Problem 5

For the following problems, perform the divisions. $$ \frac{y^{3}+3 y^{2}+4}{y+2} $$

Short Answer

Expert verified

Question: Perform polynomial division with the dividend $y^3 + 3y^2 + 4$ and the divisor $y + 2$. Answer: The result of the division is $y^2 + \frac{y^2+4}{y+2}$.

Step by step solution

Divide the highest degree term of the dividend by the highest degree term of the divisor

Divide $y^3$ by $y$ to get $y^2$. This will be the first term of our quotient.

Multiply the divisor by the result from step 1

Now, multiply the divisor, $y + 2$, by the result from step 1, $y^2$. We have $(y+2)(y^2) = y^3 + 2y^2$.

Subtract the product from step 2 from the dividend

Subtract the product from step 2, $y^3 + 2y^2$, from the dividend, $y^3 + 3y^2 + 4$. We get $(y^3 + 3y^2 + 4) - (y^3 + 2y^2) = y^2 + 4$.

Check if the degree of the new dividend is less than the degree of the divisor

The degree of the new dividend, $y^2 + 4$, is 2 and the degree of the divisor, $y + 2$, is 1. Since the degree of the new dividend is not less than the degree of the divisor, we cannot apply step 4, and division is complete. The quotient we obtained is $y^2$ and the remainder is $y^2 + 4$. Therefore, the result of the division can be expressed as $$ \frac{y^3+3y^2+4}{y+2} = y^2 + \frac{y^2+4}{y+2}. $$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Long Division of Polynomials

Just like long division in arithmetic, long division of polynomials involves a few systematic steps to reach a result. It's a method used to divide a polynomial by another polynomial of the same or a lower degree. The process might seem tricky, but with practice, it becomes much easier.

Here鈥檚 how it works in a simple form:

Divide: Look at the highest degree term of the dividend. Divide it by the highest degree term of the divisor to get the first term of the quotient.
Multiply: Multiply the entire divisor by the term obtained in the previous step.
Subtract: Subtract the result of the multiplication from the original dividend.
Repeat: Use the result of the subtraction as your new dividend. Repeat these steps until the dividend's degree is less than the divisor鈥檚 degree.

In the given exercise, this method allowed us to divide the polynomial $y^3 + 3y^2 + 4$ by $y + 2$, resulting in the quotient $y^2$, with a remainder.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Understanding these expressions is key in performing operations like addition, subtraction, multiplication, and division with polynomials.

For example, in the exercise:

The dividend is $y^3 + 3y^2 + 4$, which is an algebraic expression of polynomial type.
The divisor is $y + 2$, another polynomial expression.

Both are written in terms of the variable $y$. Polynomials express relationships and develop into more complex expressions by simply performing arithmetic with the terms. Mastering the manipulation of algebraic expressions is foundational for algebra.

Remainder Theorem

The Remainder Theorem is a useful tool when dividing polynomials. It tells us that if a polynomial $f(x)$ is divided by $x - a$, the remainder of this division is $f(a)$. This theorem provides a quick way to evaluate the remainder of a polynomial division without performing complete long division.

In the division we worked through, with $y + 2$ as the divisor, if we wanted to find the remainder directly, we would plug $-2$ into the dividend $y^3 + 3y^2 + 4$ and evaluate. However, since our divisor isn't in the form $x - a$, traditional polynomial long division was needed. Still, when applicable, the Remainder Theorem helps simplify the problem.

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91影视

For the following problems, perform the divisions. $$ \frac{y^{3}+3 y^{2}+4}{y+2} $$

Short Answer

Step by step solution

Divide the highest degree term of the dividend by the highest degree term of the divisor

Multiply the divisor by the result from step 1

Subtract the product from step 2 from the dividend

Check if the degree of the new dividend is less than the degree of the divisor

Key Concepts

Long Division of Polynomials

Algebraic Expressions

Remainder Theorem

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