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Magazine Chapter 6: Problem 62
Simplify. \(\frac{3 x^{4}+x^{3}-8 x^{2}+10 x-3}{3 x-2}\)
Short Answer
Expert verified
Divide using polynomial long division to simplify: \(x^3 + \frac{1}{3}x^2 - \frac{22}{9}x + \frac{40}{27} + \frac{49}{27(3x - 2)}\).
Step by step solution
01
Set Up Polynomial Division
To simplify the expression \(\frac{3x^4 + x^3 - 8x^2 + 10x - 3}{3x - 2}\), we need to perform polynomial long division. Start by dividing the leading term of the numerator, \(3x^4\), by the leading term of the denominator, \(3x\). So, \(\frac{3x^4}{3x} = x^3\). This \(x^3\) is the first term of the quotient.
02
Multiply and Subtract
Multiply the entire divisor \((3x - 2)\) by the first term of the quotient, \(x^3\), which gives \(3x^4 - 2x^3\). Subtract this product from the original polynomial \(3x^4 + x^3 - 8x^2 + 10x - 3\) to get \((x^3 + 0x^2 + 10x - 3) - (2x^3) = 3x^3 - 8x^2 + 10x - 3\).
03
Repeat the Division Process
The new polynomial expression is \(x^3 - 8x^2 + 10x - 3\). Divide the leading new term, \(x^3\), by the divisor's leading term, \(3x\), which gives \(\frac{x^3}{3x} = \frac{1}{3}x^2\). Multiply \(3x - 2\) by \(\frac{1}{3}x^2\) resulting in \(x^3 - \frac{2}{3}x^2\), and subtract from the current dividend to yield \(\left(-\frac{22}{3}x^2 + 10x - 3\right)\).
04
Continue the Process
Repeat by dividing \(-\frac{22}{3}x^2\) by \(3x\), producing \(-\frac{22}{9}x\). Multiply \(3x - 2\) by \(-\frac{22}{9}x\) to get \(-\frac{66}{9}x + \frac{44}{9}\). Subtract this from the running dividend: \((10x - 3) - \left(-\frac{66}{9}x + \frac{44}{9}\right) = \left(\frac{40}{9}x - \frac{71}{9}\right)\).
05
Final Division and Subtraction
Divide \(\frac{40}{9}x\) by \(3x\) to get \(\frac{40}{27}\). Multiply \(3x - 2\) by \(\frac{40}{27}\) to yield \(\frac{40}{9}x - \frac{80}{27}\). Subtract this from \(\frac{40}{9}x - \frac{71}{9}\) to get the remainder: \(\left(\frac{40}{9}x - \frac{71}{9}\right) - \left(\frac{40}{9}x - \frac{80}{27}\right) = \frac{49}{27}\).
06
Write the Simplified Form
The simplified form of the original expression is the quotient \(x^3 + \frac{1}{3}x^2 - \frac{22}{9}x + \frac{40}{27}\) with a remainder of \(\frac{49}{27}\), which can be written as: \[x^3 + \frac{1}{3}x^2 - \frac{22}{9}x + \frac{40}{27} + \frac{49}{27(3x - 2)}\].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Simplifying Algebraic Expressions
Simplifying algebraic expressions involves breaking down a complex mathematical expression into a more digestible or manageable form. In polynomial division, the aim is to express the polynomial in a simplified quotient-remainder form. Simplification makes it easier to understand the nature of the polynomial and find any patterns or symmetries.
To simplify an algebraic expression using polynomial division:
To simplify an algebraic expression using polynomial division:
- Identify the leading terms in the numerator (the dividend) and the denominator (the divisor).
- Perform division on the leading terms to find the first term of the quotient.
- Multiply the entire divisor by this quotient term and subtract the result from your original polynomial.
- Repeat this process using the new polynomial formed by subtraction until the remaining degree is less than that of the divisor.
Remainder Theorem
The Remainder Theorem is a powerful tool in algebra that connects polynomial division with evaluating polynomials. When you divide a polynomial by a linear divisor of the form \(x - r\), the remainder of this division is equal to the value of the polynomial at \(x = r\).
This theorem is particularly handy when you want to quickly determine if something is a factor of a polynomial as a zero remainder indicates divisibility. Therefore, while performing the division:
This theorem is particularly handy when you want to quickly determine if something is a factor of a polynomial as a zero remainder indicates divisibility. Therefore, while performing the division:
- Once the polynomial is divided, check the final remainder.
- If the remainder is zero, the divisor is a factor of the polynomial.
- If not, the remainder can be used as described, or added back into the simplified form of the polynomial.
Polynomial Division Steps
Polynomial long division is a methodical technique similar to numerical long division, but applied to polynomials. Each step helps in breaking down a complex polynomial into simpler parts.
Follow these steps for polynomial division:
Follow these steps for polynomial division:
- Set Up the Division: Write the dividend (numerator) and the divisor side by side, just like in regular division.
- Divide the Leading Terms: Determine how many times the leading term of the divisor goes into the leading term of the dividend to get the first term of the quotient.
- Multiply and Subtract: Multiply the entire divisor by the quotient term obtained and subtract this product from the original polynomial.
- Bring Down the Next Term: If necessary, bring down the next term from the dividend to form a new polynomial and repeat the process.
- Continue Until Remainder is Zero or of Lesser Degree: Keep dividing, multiplying, and subtracting until you achieve a remainder that has a lesser degree than the divisor.
