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Magazine Chapter 2: Problem 58
Divide the polynomials by either long division or synthetic division. $$\left(x^{3}-8\right) \div\left(x^{2}-2\right)$$
Short Answer
Expert verified
Quotient: \( x \) with Remainder: \( 2x - 8 \)
Step by step solution
01
Identify the Division Method
We are given the polynomials \( x^3 - 8 \) as the dividend and \( x^2 - 2 \) as the divisor. Since the divisor has a higher degree than a simple linear polynomial, it is best to utilize long division to solve this exercise.
02
Set Up Long Division
Write the dividend \( x^3 - 0x^2 + 0x - 8 \) under the division symbol and the divisor \( x^2 - 2 \) outside the division symbol. Remember to include all coefficients, even zeros, for placeholders of missing terms.
03
Divide the Leading Terms
Divide the leading term of the dividend \( x^3 \) by the leading term of the divisor \( x^2 \) to get the first term of the quotient: \( x^3 / x^2 = x \).
04
First Multiplication
Multiply the entire divisor \( x^2 - 2 \) by the first term of the quotient \( x \), which gives \( x(x^2 - 2) = x^3 - 2x \). Write this underneath the dividend and subtract.
05
First Subtraction
Subtract \( x^3 - 2x \) from \( x^3 - 0x^2 + 0x - 8 \), resulting in \( 0x^2 + 2x - 8 \). Drop down the next term to continue the division.
06
Divide the New Leading Terms
Now divide the new leading term \( 0x^2 \) by \( x^2 \) to get \( 0 \). As this doesn't contribute, move on to divide the term based on the next coefficient: Divide \( 2x \) by \( x^2 \), which results in \( 0 \). Since these do not add any new significant terms to the quotient, we proceed directly to consider what remains.
07
Analyzing the Remainder
The remainder now stands at \( 2x - 8 \). Since there is no way to further divide by the divisor in a significant manner (as expected, because we've exhausted degrees/can go no further in the quotient), this makes \( 2x - 8 \) the final remainder.
08
Complete the Division
Our division process results in the quotient \( x \) with the remainder \( 2x - 8 \). The division can be expressed as: \( x + \frac{2x - 8}{x^2 - 2} \)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Long Division
Long division is a systematic process used to divide polynomials and is very similar to dividing numbers. When the divisor is a polynomial of degree two or higher, long division is preferred. It provides a clear framework for understanding the division of polynomials step-by-step.
Here鈥檚 how long division of polynomials works:
Here鈥檚 how long division of polynomials works:
- Write the dividend under the division bracket and the divisor to the left. Include all terms of the dividend even if they have a coefficient of zero. This ensures each column aligns with the correct powers of the variable.
- Identify the first term of the dividend and divide it by the first term of the divisor. This provides the first term of the quotient.
- Multiply this term by the entire divisor and subtract from the dividend to find the new remaining polynomial.
- Continue the process by repeating these steps until the degree of the remaining polynomial is less than the degree of the divisor.
- The remaining polynomial is the remainder of the division.
Synthetic Division
Synthetic division offers a streamlined method primarily used when dividing a polynomial by a linear divisor, specifically when the divisor is of the form \(x - c\) where \(c\) is a constant.
Unlike long division that requires handling each term separately, synthetic division simplifies this process significantly:
Unlike long division that requires handling each term separately, synthetic division simplifies this process significantly:
- Write down the coefficients of the polynomial you want to divide. You do not have to include the variables and exponents with synthetic division.
- Place the constant \(c\) from the divisor \(x - c\) to the side. This constant is what your entire division rests upon.
- Bring the leading coefficient straight down as it is your starting point for the calculations.
- Multiply the leading coefficient by \(c\), then add it to the next coefficient in your list.
- Repeat this process for each coefficient. The last result obtained (if any) will be the remainder.
Polynomial Remainder
The process of dividing polynomials isn't always perfectly divisible, meaning you might end up with a remainder, which is a polynomial of a lower degree than the divisor.
The remainder theorem provides useful insight in this context:
The remainder theorem provides useful insight in this context:
- The polynomial division produces a quotient and a remainder.
- The remainder can be expressed as part of the division result like this: the original polynomial is equal to the divisor times the quotient plus the remainder. Mathematically, \[P(x) = D(x)Q(x) + R(x)\]
- If you evaluate the original polynomial function at the divisor's zero (\(x = c\), assuming a linear divisor \(x - c\)), it'll equate to the remainder.
