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Chapter 3: Problem 18
Use long division to divide. $$\frac{12 x^{4}-4 x^{3}+13 x^{2}+2 x+1}{3 x^{2}-x+4}$$
Short Answer
Expert verified
The quotient is \(4x^2 - 1\) with a remainder of \(x + 5\).
Step by step solution
01
- Set up the division
Write the dividend, \(12x^4 - 4x^3 + 13x^2 + 2x + 1\), and the divisor, \(3x^2 - x + 4\), for long division. Place them as follows: \[ \text{Dividend} \bigg\backslash \text{Divisor} \ \frac{12 x^4-4 x^3+13 x^2+2 x+1}{3 x^{2}-x+4} \]
02
- Divide the leading terms
Divide the first term of the dividend, \(12x^4\), by the first term of the divisor, \(3x^2\), giving \(4x^2\). Write this quotient term above the long division bar.
03
- Multiply and subtract
Multiply \(4x^2\) by the entire divisor \(3x^2-x+4\) to get \(12x^4 - 4x^3 + 16x^2\). Subtract this from the dividend: \[ (12x^4 - 4x^3 + 13x^2 + 2x + 1) - (12x^4 - 4x^3 + 16x^2) = -3x^2 + 2x + 1 \]
04
- Repeat the process
Bring down the next term and repeat the division process. Divide \(-3x^2\) by \(3x^2\) to get \(-1\). Multiply \(-1\) by \(3x^2 - x + 4\) yielding \(-3x^2 + x - 4\). Subtract this result from the current dividend to get: \[ (-3x^2 + 2x + 1) - (-3x^2 + x - 4) = x + 5 \]
05
- Finalize the quotient and remainder
Since \(x + 5\) has a lower degree than the divisor \(3x^2 - x + 4\), it becomes the remainder of our division. Thus, the quotient is \(4x^2 - 1\) and the remainder is \(x + 5\).
06
- Express the final answer
The result of our long division can be written as: \[ \frac{12 x^{4}-4 x^{3}+13 x^{2}+2 x+1}{3 x^{2}-x+4} = 4x^2 - 1 + \frac{x+5}{3x^2 - x + 4} \]
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial Division
To divide polynomials, we use a method similar to long division with numbers. Polynomial division involves dividing a polynomial (the dividend) by another polynomial (the divisor). Each term in the dividend is divided by the leading term of the divisor.
This method helps simplify complex expressions and find factors of polynomials.
Let’s start with understanding the dividend and divisor in our problem:
Dividend: \(12x^4 - 4x^3 + 13x^2 + 2x + 1\)
Divisor: \(3x^2 - x + 4\)
We'll use these polynomials to illustrate each step in the division process.
Quotient and Remainder
The quotient and remainder are essential concepts in polynomial division.
The quotient is the result obtained by dividing the leading terms in each step. In our example, the first term divided is \(12x^4\) by \(3x^2\), resulting in \(4x^2\), which was our initial quotient term.
At each step, we subtract the product of the divisor and this quotient term from the dividend. The process is repeated for the next terms until the degree of the remaining polynomial (remainder) is less than the degree of the divisor.
In our case, after division and subtraction, we get a final quotient of \(4x^2 - 1\) and a remainder of \(x + 5\).
Our division result can be written as: \[ \frac{12 x^{4}-4 x^{3}+13 x^{2}+2 x+1}{3 x^{2}-x+4} = 4 x^2 - 1 + \frac{x+5}{3 x^2-x + 4} \]
Algebraic Expressions
Algebraic expressions are combinations of variables, coefficients, and operators. They are central to solving equations and simplifying terms throughout polynomial division.
Our dividend and divisor are examples of algebraic expressions. The dividend \(12x^4 - 4x^3 + 13x^2 + 2x + 1\), and divisor \(3x^2 - x + 4\) are polynomials of varying degrees.
Understanding how to manipulate and simplify these expressions through steps such as addition, subtraction, multiplication, and division is key to working efficiently with polynomials.
These concepts are used throughout fields like engineering, computer science, and economics, where polynomial expressions come in handy to solve real-world problems.
Step-by-Step Problem-Solving
Polynomial division can be daunting, but breaking it down step-by-step makes it manageable.
Here's a quick recap of our structured approach:
- Step 1: Set Up
Write the dividend and divisor ready for division. - Step 2: Divide Leading Terms
Divide the leading term of the dividend by the leading term of the divisor. - Step 3: Multiply and Subtract
Multiply the quotient term by the entire divisor and subtract from the dividend. - Step 4: Repeat
Bring down the next term and repeat the process until the degree of the remainder is less than the divisor. - Step 5: Finalize
Recognize the final quotient and remainder, and express the division result in the proper format.
By following these steps, you can solve complex polynomial divisions efficiently and accurately.
Remember, practice makes perfect!
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$... [FREE SOLUTION] (2024)](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mP8/h8AAvMB+NzmkbcAAAAASUVORK5CYII=)