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Chapter 6: Problem 39

Divide and check. $$ \left(3 x^{4}+x^{3}-8 x^{2}-3 x-3\right) \div\left(x^{2}-3\right) $$

Short Answer

Expert verified
Quotient: \(3x^2 + x + 4\), Remainder: \(15\)

Step by step solution

01

- Setup the Problem for Long Division

We'll perform polynomial long division. First, write down the dividend \(\text{{3}}x^4 + x^3 - 8x^2 - 3x - 3\) and the divisor \(x^2 - 3\).
02

- Divide the Leading Terms

Divide the leading term of the dividend (\(3x^4\)) by the leading term of the divisor (\(x^2\)). The result is \(3x^2\). Write \(3x^2\) above the division line.
03

- Multiply and Subtract

Multiply \(3x^2\) by the divisor \(x^2 - 3\) to get \(3x^4 - 9x^2\). Subtract \(3x^4 - 9x^2\) from the dividend. This gives the new polynomial \(x^3 + x^2 - 3x - 3\).
04

- Repeat Division and Subtraction

Next, divide the new leading term \(x^3\) by \(x^2\) to get \(x\). Multiply \(x\) by the divisor to get \(x^3 - 3x\). Subtract this from the new polynomial to get \(4x^2 +3\).
05

- Final Division and Subtraction

Finally, divide the new leading term \(4x^2\) by \(x^2\) to get \(4\). Multiply \(4\) by the divisor to get \(4x^2 - 12\). Subtract this from the polynomial to get the remainder \(15\).
06

- Write the Solution

Combining all the terms, the quotient is \(3x^2 + x + 4\) and the remainder is \(15\). The full answer can be written as \((3x^2 + x + 4) + \frac{15}{x^2-3}\).

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Key Concepts

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

Algebraic Division
Algebraic division is a fundamental concept you need to understand when dividing polynomials. It's similar to numbers' long division but involves variables. In our problem, we divide the polynomial \(3x^4 + x^3 - 8x^2 - 3x - 3\) by the polynomial \(x^2 - 3\). The main thing you need to grasp is that we repeatedly focus on the leading (first) terms to simplify step-by-step.
Division Algorithm
The division algorithm helps you understand how polynomial division works. It's a step-by-step method to break down polynomials. Here's a simplified guide to follow:
1. **Divide the Leading Terms**: Start by dividing the first term of the dividend by the first term of the divisor.
2. **Multiply and Subtract**: Multiply the result by the entire divisor, then subtract this product from the polynomial.
3. **Repeat**: Repeat the process with the new leading term generated after subtraction.
Using this approach, you handle polynomial terms systematically, reducing errors and attaining a clear quotient.
Polynomial Remainder
After performing the division steps, you might get a smaller polynomial or number that remains, called the **remainder**. In our exercise, the process leaves a remainder of \(15\). Understanding remainders is crucial because they show how the original polynomial isn't perfectly divisible by the divisor. The result can be written in the form of a quotient and remainder as \( (3x^2 + x + 4) + \frac{15}{x^2 - 3} \), clearly combining both the quotient and remainder in the final solution.

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