Problem 46 - Algebra for College Students

Question

Perform the indicated divisions. $$ \left(3 n^{4}+n^{3}-7 n^{2}-2 n+2\right) \div\left(n^{2}-2\right) $$

Step-by-Step Solution

Answer

Quotient is \( 3n^2 + n - 1 \) with remainder 2.

Step 1: Set Up the Long Division

To divide the polynomial \( 3n^4 + n^3 - 7n^2 - 2n + 2 \) by \( n^2 - 2 \), set up the division by writing the dividend \( 3n^4 + n^3 - 7n^2 - 2n + 2 \) under the long division symbol, and the divisor \( n^2 - 2 \) outside it.

Step 2: Divide the Leading Terms

Divide the leading term of the dividend, \( 3n^4 \), by the leading term of the divisor, \( n^2 \), which gives \( 3n^2 \). Write \( 3n^2 \) above the long division bar.

Step 3: Multiply and Subtract

Multiply \( 3n^2 \) by the entire divisor \( n^2 - 2 \) to get \( 3n^4 - 6n^2 \). Write this result below the original dividend and subtract to find the remainder: \[(3n^4 + n^3 - 7n^2 - 2n + 2) - (3n^4 - 6n^2) = n^3 - n^2 - 2n + 2.\]

Step 4: Repeat the Process

Divide the leading term of the new dividend \( n^3 \) by the leading term of the divisor \( n^2 \), which gives \( n \). Place \( n \) above the long division bar next to \( 3n^2 \).

Step 5: Multiply and Subtract Again

Multiply \( n \) by the divisor \( n^2 - 2 \) to get \( n^3 - 2n \). Subtract it from the current dividend: \[(n^3 - n^2 - 2n + 2) - (n^3 - 2n) = -n^2 + 4.\]

Step 6: Final Division

Now divide \( -n^2 \) by \( n^2 \) which results in \( -1 \). Place \( -1 \) above the long division bar next to \( 3n^2 + n \).

Step 7: Multiply and Subtract Final Time

Multiply \( -1 \) by \( n^2 - 2 \) to obtain \( -n^2 + 2 \). Subtract this from the remainder: \[(-n^2 + 4) - (-n^2 + 2) = 2.\]

Step 8: Write Down the Result

Since \( 2 \) is less than the degree of \( n^2 - 2 \), it is the remainder. The quotient is \( 3n^2 + n - 1 \) with a remainder of \( 2 \). Thus, the result of the division is:\[3n^2 + n - 1 + \frac{2}{n^2 - 2}.\]

Key Concepts

polynomial division

Polynomial division is similar to long division we use in arithmetic, but instead, we deal with polynomials. It's a systematic method to divide one polynomial by another, usually a simpler one. This technique allows us to find both the quotient and the remainder of division. In this exercise, we divided a higher degree polynomial, \(3n^4 + n^3 - 7n^2 - 2n + 2\), by a simpler polynomial, \(n^2 - 2\).

Here's how you can proceed with polynomial division:

• Begin by dividing the highest degree term of the dividend by the highest degree term of the divisor. This gives the first term of the quotient.
• Multiply the entire divisor by this quotient term, and subtract the result from the dividend.
• Repeat the process with the new dividend formed by the remainder until the remainder's degree is less than that of the divisor.
• The expression formed above the division symbol is the quotient, while any remainder left is part of the final answer.

Understanding each step ensures that the whole process goes smoothly, even though it initially might look a bit intimidating.

algebraic expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They represent relationships and can be simplified or manipulated to find various solutions. In our division problem, expressions like \(3n^4 + n^3 - 7n^2 - 2n + 2\) and \(n^2 - 2\) are algebraic expressions. In algebra, handling these expressions effectively is key to solving equations and other problems.

When we perform operations like division on algebraic expressions, we're actually simplifying complex relationships among variables and constants.

• Operations on algebraic expressions follow rules similar to those for numbers, such as the distributive, associative, and commutative laws.
• It's important to combine like terms and keep track of variable exponents during calculations.
• Being accurate with each step helps prevent errors and allows you to achieve the correct solution effectively.

By mastering algebraic expressions, you unlock a powerful tool to work through various mathematical problems.

rational expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. These expressions are similar to numerical fractions, and they follow similar rules when it comes to operations such as addition, subtraction, multiplication, and division. In the final answer from our division exercise, we ended up with the following rational expression: \( \frac{2}{n^2 - 2} \).

When working with rational expressions, it's crucial to:

• Identify and factor both the numerator and the denominator where possible, which can help simplify the expression.
• Keep track of any restrictions on the variables; for example, values that might make the denominator zero are not allowed.
• Understand how to add, subtract, multiply, and divide these expressions by applying similar principles used in simplifying fractions.

A solid grasp of rational expressions enables you to solve complex problems involving polynomial calculations and relations effectively.