Problem 90
Question
Divide. $$\left(-x^{2}-1\right) \div(3 x-9)$$
Step-by-Step Solution
Verified Answer
The quotient is \(-\frac{1}{3}x - 1 + \frac{-10}{3(x-3)}\)."
1Step 1: Identify Terms
First, let's identify the terms in the dividend and divisor. The dividend is \(-x^2-1\) and the divisor is \(3x-9\).
2Step 2: Simplify the Divisor
Notice that the divisor \(3x-9\) can be factored. It can be rewritten as \(3(x-3)\) by factoring out the greatest common factor, which is 3.
3Step 3: Set Up the Division Expression
The division can now be written as: \(\frac{-x^2-1}{3(x-3)}\).
4Step 4: Attempt to Simplify the Division
To simplify \(\frac{-x^2-1}{3(x-3)}\), check if any terms factor out or can be simplified. However, \(-x^2-1\) cannot be factored further or cancelled with terms in \(3(x-3)\).
5Step 5: Perform Polynomial Long Division
Since \(-x^2-1\) and \(3(x-3)\) cannot be simplified further, perform polynomial long division. Begin by dividing the leading term of the dividend \(-x^2\) by the leading term of the divisor \(3x\), which is \(-\frac{1}{3}x\). Multiply \(-\frac{1}{3}x\) by \(3x-9\) to get \(-x^2 + 3x\). Subtract this from \(-x^2 - 1\) resulting in \(-3x-1\).
6Step 6: Continue Division
Divide the new leading term \(-3x\) by \(3x\), which gives \(-1\). Multiply \(-1\) by \(3x-9\) to get \(-3x+9\). Subtract this from \(-3x-1\), resulting in \(-10\).
7Step 7: Write Final Expression
The result of the division is \(-\frac{1}{3}x - 1 + \frac{-10}{3(x-3)}\). This is the quotient with the remainder expressed as a fraction over the original divisor.
Key Concepts
FactoringSimplifying Algebraic ExpressionsPolynomial Long Division
Factoring
Factoring is a fundamental concept in algebra that involves breaking down an expression into a product of simpler expressions. This process is much like breaking down a number into its prime factors. In the context of polynomial division, factoring helps simplify expressions, making the division process easier to handle.
For instance, in the provided exercise, the divisor is initially given as \(3x - 9\). Recognizing that 3 is a common factor, we can factor the divisor to \(3(x - 3)\). This transformation reveals that both terms in the original divisor share a common factor, simplifying the structure.
Factoring can significantly change the approach to solving an exercise as it might expose opportunities for cancellation or simplify the problem into smaller, manageable parts. Even if direct simplification isn't possible, such factorization aids in setting up the problem for polynomial long division, as it helps in clearly identifying the terms involved.
For instance, in the provided exercise, the divisor is initially given as \(3x - 9\). Recognizing that 3 is a common factor, we can factor the divisor to \(3(x - 3)\). This transformation reveals that both terms in the original divisor share a common factor, simplifying the structure.
Factoring can significantly change the approach to solving an exercise as it might expose opportunities for cancellation or simplify the problem into smaller, manageable parts. Even if direct simplification isn't possible, such factorization aids in setting up the problem for polynomial long division, as it helps in clearly identifying the terms involved.
Simplifying Algebraic Expressions
Simplifying algebraic expressions involves reducing them to a form that is easier to interpret and work with. It's about making expressions as straightforward as possible without changing their value. This often includes combining like terms, factoring, and canceling common factors.
In this exercise, the attempt to simplify \(\frac{-x^2-1}{3(x-3)}\) involved checking whether the dividend \(-x^2 - 1\) could be factored or simplified further. Unfortunately, in this specific case, \(-x^2 - 1\) doesn't factor nicely, meaning no common factors with the divisor could be canceled out.
A simplified expression is generally more tractable for both analysis and further algebraic operations like addition, subtraction, or division. Even though further simplification wasn't possible in this case, preparing the expression by attempting to simplify guides the succeeding operations – in this scenario, transitioning to polynomial long division.
In this exercise, the attempt to simplify \(\frac{-x^2-1}{3(x-3)}\) involved checking whether the dividend \(-x^2 - 1\) could be factored or simplified further. Unfortunately, in this specific case, \(-x^2 - 1\) doesn't factor nicely, meaning no common factors with the divisor could be canceled out.
A simplified expression is generally more tractable for both analysis and further algebraic operations like addition, subtraction, or division. Even though further simplification wasn't possible in this case, preparing the expression by attempting to simplify guides the succeeding operations – in this scenario, transitioning to polynomial long division.
Polynomial Long Division
Polynomial long division is a method used to divide one polynomial by another, similar to the long division of numbers. It is particularly useful when polynomial expressions cannot be simplified further by factorization alone. In this exercise, polynomial long division was essential to find the quotient and remainder.
The process started by dividing the leading term of the dividend \(-x^2\) by the leading term of the divisor \(3x\), resulting in \(-\frac{1}{3}x\). After multiplying \(-\frac{1}{3}x\) by the entire divisor \(3x - 9\), you subtract it from the original dividend. This leads to a new expression \(-3x - 1\).
The division continues by repeating these steps, handling each term systematically. Each step refines the polynomial further, until a remainder that cannot be divided by the divisor's leading term emerges. In this solution, the final expression becomes \(-\frac{1}{3}x - 1 + \frac{-10}{3(x-3)}\), combining the quotient and any remainder over the original divisor.
The process started by dividing the leading term of the dividend \(-x^2\) by the leading term of the divisor \(3x\), resulting in \(-\frac{1}{3}x\). After multiplying \(-\frac{1}{3}x\) by the entire divisor \(3x - 9\), you subtract it from the original dividend. This leads to a new expression \(-3x - 1\).
The division continues by repeating these steps, handling each term systematically. Each step refines the polynomial further, until a remainder that cannot be divided by the divisor's leading term emerges. In this solution, the final expression becomes \(-\frac{1}{3}x - 1 + \frac{-10}{3(x-3)}\), combining the quotient and any remainder over the original divisor.
- Long division breaks down complex problems step-by-step.
- It’s especially helpful when factoring doesn't provide solutions.
- Enables the division of higher-degree polynomials by simpler terms.
Other exercises in this chapter
Problem 89
Find the conjugate of each number. $$5-3 i$$
View solution Problem 90
Use the boundedness theorem to show that the real zeros of \(P(x)\) satisfy the given conditions. \(P(x)=x^{5}-3 x^{3}+x+2\); no real zero less than \(-3\)
View solution Problem 90
Find the conjugate of each number. $$-3+i$$
View solution Problem 91
Divide. $$\left(x^{3}-x^{2}+1\right) \div\left(2 x^{2}-1\right)$$
View solution