Problem 70
Question
Simplify the rational expression. $$\frac{3 x^{3}-5 x^{2}-34 x+24}{3 x-2}$$
Step-by-Step Solution
Verified Answer
\(x^{2}-x-\frac{32}{3}+\frac{8}{3x-2}\)
1Step 1: Polynomial Division
Begin by performing polynomial division on the numerator by the denominator. Divide the first term of the numerator by the first term of the denominator to find the first term of the quotient. So, \(3x^3\) divided by \(3x\) gives \(x^2\).
2Step 2: Multiply and Subtract
Multiply \(x^2\) by every term in the divisor \(3x - 2\) and subtract the resulting expression from the original numerator: \(3x^3 - 5x^2 -34x + 24 - (x^2*(3x - 2)) = 3x^3 - 5x^2 - 34x + 24 - (3x^3 - 2x^2) = -3x^2 - 34x + 24.
3Step 3: Repeat the Process
Repeat steps 1 and 2 for \(-3x^2 - 34x + 24\). \(-3x^2\) divided by \(3x\) is \(-x\). We multiply \(-x\) by every term in the divisor and subtract from \(-3x^2 - 34x + 24\) to give a new dividend: \(-3x^2 - 34x + 24 - (-x*(3x - 2)) = -3x^2 - 34x + 24 - (-3x^2 + 2x) = -32x + 24.
4Step 4: Repeat again
Again, repeat steps 1 and 2 for \(-32x + 24\). \(-32x\) divided by \(3x\) is \(-\frac{32}{3}\). We multiply \(-\frac{32}{3}\) by every term in the divisor and subtract from \(-32x + 24\) to give the remainder: \(-32x + 24 -(-\frac{32}{3}*(3x - 2)) = -32x + 24 - (-32x + \frac{64}{3}) = \frac{8}{3}.
5Step 5: Write the final result
Write down the quotient and remainder together to express the simplified rational expression. The desired simplification of the given expression is obtained by adding the remainder over the divisor to the quotient. So, the answer is \(x^2 - x - \frac{32}{3} + \frac{8}{3x - 2}\).
Key Concepts
Rational ExpressionsSimplificationQuotient and Remainder
Rational Expressions
Understanding rational expressions is essential when working with polynomial division. Essentially, a rational expression is like a fraction but with polynomials in the numerator and the denominator instead of simple numbers. For example, the expression \(\frac{3x^3 - 5x^2 - 34x + 24}{3x-2}\) is a rational expression because both the numerator and the denominator are polynomials.
Just like with numerical fractions, there are rules for operating with rational expressions. We can add, subtract, multiply, and divide them, but each operation follows its own set of principles. When dealing with division in rational expressions, it's crucial to ensure the denominator is never zero. In our given problem, the divisor is \(3x - 2\), and we must remember that when \(3x = 2\), the expression is undefined, meaning \(x\) cannot be \(\frac{2}{3}\).
Sometimes rational expressions can be simplified if the numerator and the denominator share common factors. The main goal, when simplifying, is to reduce the expression to its simplest form by canceling out these factors. However, this simplification often requires performing polynomial division first, as illustrated in the given exercise.
Just like with numerical fractions, there are rules for operating with rational expressions. We can add, subtract, multiply, and divide them, but each operation follows its own set of principles. When dealing with division in rational expressions, it's crucial to ensure the denominator is never zero. In our given problem, the divisor is \(3x - 2\), and we must remember that when \(3x = 2\), the expression is undefined, meaning \(x\) cannot be \(\frac{2}{3}\).
Sometimes rational expressions can be simplified if the numerator and the denominator share common factors. The main goal, when simplifying, is to reduce the expression to its simplest form by canceling out these factors. However, this simplification often requires performing polynomial division first, as illustrated in the given exercise.
Simplification
Simplification is at the heart of managing rational expressions, and it involves reducing them to their simplest form. This process often begins with dividing the polynomials through polynomial division, a systematic procedure similar to long division in arithmetic.
In the exercise, to simplify \(\frac{3x^3 - 5x^2 - 34x + 24}{3x-2}\), we perform polynomial division. By breaking down each term and performing polynomial division step by step, we uncover both a clear quotient and a remainder.
Simplifying involves:
In the exercise, to simplify \(\frac{3x^3 - 5x^2 - 34x + 24}{3x-2}\), we perform polynomial division. By breaking down each term and performing polynomial division step by step, we uncover both a clear quotient and a remainder.
Simplifying involves:
- Dividing the leading terms of the numerator and denominator.
- Multiplying the result back with the entire divisor and subtracting from the original polynomial.
- Repeating the above steps until we reduce the terms further and further.
Quotient and Remainder
In the polynomial division process, obtaining the quotient and remainder is a key part of simplifying rational expressions.
The quotient is what you get when you divide the numerator by the denominator, similar to how division works with numbers. It tells you how many times a polynomial (the divisor) fits into another polynomial (the dividend). In our example, after performing the division, the quotient is \(x^2 - x - \frac{32}{3}\).
On the other hand, the remainder is what is left over after the division process—sort of like what remains in your school lunchbox after you give out cookies to your friends. In this example, the remainder is \(\frac{8}{3x - 2}\). In mathematical expressions, this remainder is often added back to the quotient as a fraction over the divisor.
When writing down the final result of simplifying a rational expression, you combine both the quotient and remainder. The expression becomes the sum of the quotient and the remainder part over the original divisor, which in our case completes as \(x^2 - x - \frac{32}{3} + \frac{8}{3x - 2}\). Therefore, mastering these concepts helps in understanding how polynomials interact when divided and how to express them elegantly.
The quotient is what you get when you divide the numerator by the denominator, similar to how division works with numbers. It tells you how many times a polynomial (the divisor) fits into another polynomial (the dividend). In our example, after performing the division, the quotient is \(x^2 - x - \frac{32}{3}\).
On the other hand, the remainder is what is left over after the division process—sort of like what remains in your school lunchbox after you give out cookies to your friends. In this example, the remainder is \(\frac{8}{3x - 2}\). In mathematical expressions, this remainder is often added back to the quotient as a fraction over the divisor.
When writing down the final result of simplifying a rational expression, you combine both the quotient and remainder. The expression becomes the sum of the quotient and the remainder part over the original divisor, which in our case completes as \(x^2 - x - \frac{32}{3} + \frac{8}{3x - 2}\). Therefore, mastering these concepts helps in understanding how polynomials interact when divided and how to express them elegantly.
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