Solved on Feb 10, 2024

Simplify the expression $(x^{2}+5) \div(x-1)$ and select the correct answer.

STEP 1

Assumptions
1. We are given a rational expression $(x^{2}+5) \div (x-1)$ .
2. We need to perform polynomial long division or simplify the expression to match one of the given answer choices.

STEP 2

Begin the polynomial long division process by dividing the first term of the numerator by the first term of the denominator.
$\frac{x^{2}}{x} = x$

STEP 3

Multiply the entire divisor $(x-1)$ by the result from the previous step $x$ .
$x \cdot (x-1) = x^{2} - x$

STEP 4

Subtract the result of the multiplication from the original numerator.
$\begin{align} x^{2} + 5 - (x^{2} - x) &= x^{2} + 5 - x^{2} + x \\ &= x + 5 \end{align}$

STEP 5

Now, we have a new expression to divide by $(x-1)$ , which is $x + 5$ .

STEP 6

Divide the first term of the new expression by the first term of the divisor.
$\frac{x}{x} = 1$

STEP 7

Multiply the entire divisor $(x-1)$ by the result from the previous step $1$ .
$1 \cdot (x-1) = x - 1$

STEP 8

Subtract the result of the multiplication from the new expression.
$\begin{align} x + 5 - (x - 1) &= x + 5 - x + 1 \\ &= 6 \end{align}$

STEP 9

The result $6$ is the remainder of the division, since it cannot be divided by $(x-1)$ without resulting in a fraction.

STEP 10

Combine the quotient obtained from the division with the remainder over the divisor to write the final answer.
$\begin{align} Quotient + \frac{Remainder}{Divisor} &= x + 1 + \frac{6}{x-1} \end{align}$

STEP 11

Compare the final answer with the given options.
The correct answer is option c, which is: $(x+1)+\frac{6}{x-1}$

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Simplify the expression $(x^{2}+5) \div(x-1)$ and select the correct answer.

STEP 1

Assumptions
1. We are given a rational expression $(x^{2}+5) \div (x-1)$ .
2. We need to perform polynomial long division or simplify the expression to match one of the given answer choices.

STEP 2

Begin the polynomial long division process by dividing the first term of the numerator by the first term of the denominator.
$\frac{x^{2}}{x} = x$

STEP 3

Multiply the entire divisor $(x-1)$ by the result from the previous step $x$ .
$x \cdot (x-1) = x^{2} - x$

STEP 4

Subtract the result of the multiplication from the original numerator.
$\begin{align} x^{2} + 5 - (x^{2} - x) &= x^{2} + 5 - x^{2} + x \\ &= x + 5 \end{align}$

STEP 5

Now, we have a new expression to divide by $(x-1)$ , which is $x + 5$ .

STEP 6

Divide the first term of the new expression by the first term of the divisor.
$\frac{x}{x} = 1$

STEP 7

Multiply the entire divisor $(x-1)$ by the result from the previous step $1$ .
$1 \cdot (x-1) = x - 1$

STEP 8

Subtract the result of the multiplication from the new expression.
$\begin{align} x + 5 - (x - 1) &= x + 5 - x + 1 \\ &= 6 \end{align}$

STEP 9

The result $6$ is the remainder of the division, since it cannot be divided by $(x-1)$ without resulting in a fraction.

STEP 10

Combine the quotient obtained from the division with the remainder over the divisor to write the final answer.
$\begin{align} Quotient + \frac{Remainder}{Divisor} &= x + 1 + \frac{6}{x-1} \end{align}$

STEP 11

Compare the final answer with the given options.
The correct answer is option c, which is: $(x+1)+\frac{6}{x-1}$

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Simplify the expression (x2+5)÷(x−1)(x^{2}+5) \div(x-1)(x2+5)÷(x−1) and select the correct answer.

STEP 1

Assumptions1. We are given a rational expression (x2+5)÷(x−1)(x^{2}+5) \div (x-1)(x2+5)÷(x−1).2. We need to perform polynomial long division or simplify the expression to match one of the given answer choices.

STEP 2

Begin the polynomial long division process by dividing the first term of the numerator by the first term of the denominator.x2x=x\frac{x^{2}}{x} = xxx2​=x

STEP 3

Multiply the entire divisor (x−1)(x-1)(x−1) by the result from the previous step xxx.x⋅(x−1)=x2−xx \cdot (x-1) = x^{2} - xx⋅(x−1)=x2−x

STEP 4

Subtract the result of the multiplication from the original numerator.x2+5−(x2−x)=x2+5−x2+x=x+5 \begin{align*} x^{2} + 5 - (x^{2} - x) &= x^{2} + 5 - x^{2} + x \\ &= x + 5 \end{align*} x2+5−(x2−x)​=x2+5−x2+x=x+5​

STEP 5

Now, we have a new expression to divide by (x−1)(x-1)(x−1), which is x+5x + 5x+5.

STEP 6

Divide the first term of the new expression by the first term of the divisor.xx=1\frac{x}{x} = 1xx​=1

STEP 7

Multiply the entire divisor (x−1)(x-1)(x−1) by the result from the previous step 111.1⋅(x−1)=x−11 \cdot (x-1) = x - 11⋅(x−1)=x−1

STEP 8

Subtract the result of the multiplication from the new expression.x+5−(x−1)=x+5−x+1=6 \begin{align*} x + 5 - (x - 1) &= x + 5 - x + 1 \\ &= 6 \end{align*} x+5−(x−1)​=x+5−x+1=6​

STEP 9

The result 666 is the remainder of the division, since it cannot be divided by (x−1)(x-1)(x−1) without resulting in a fraction.

STEP 10

Combine the quotient obtained from the division with the remainder over the divisor to write the final answer.Quotient+RemainderDivisor=x+1+6x−1 \begin{align*} Quotient + \frac{Remainder}{Divisor} &= x + 1 + \frac{6}{x-1} \end{align*} Quotient+DivisorRemainder​​=x+1+x−16​​

STEP 11

Compare the final answer with the given options.The correct answer is option c, which is: (x+1)+6x−1 (x+1)+\frac{6}{x-1} (x+1)+x−16​

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Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Simplify the expression $(x^{2}+5) \div(x-1)$ and select the correct answer.

Assumptions
1. We are given a rational expression $(x^{2}+5) \div (x-1)$ .
2. We need to perform polynomial long division or simplify the expression to match one of the given answer choices.

Begin the polynomial long division process by dividing the first term of the numerator by the first term of the denominator.
$\frac{x^{2}}{x} = x$

Multiply the entire divisor $(x-1)$ by the result from the previous step $x$ .
$x \cdot (x-1) = x^{2} - x$

Subtract the result of the multiplication from the original numerator.
$\begin{align} x^{2} + 5 - (x^{2} - x) &= x^{2} + 5 - x^{2} + x \\ &= x + 5 \end{align}$

Now, we have a new expression to divide by $(x-1)$ , which is $x + 5$ .

Divide the first term of the new expression by the first term of the divisor.
$\frac{x}{x} = 1$

Multiply the entire divisor $(x-1)$ by the result from the previous step $1$ .
$1 \cdot (x-1) = x - 1$

Subtract the result of the multiplication from the new expression.
$\begin{align} x + 5 - (x - 1) &= x + 5 - x + 1 \\ &= 6 \end{align}$

The result $6$ is the remainder of the division, since it cannot be divided by $(x-1)$ without resulting in a fraction.

Combine the quotient obtained from the division with the remainder over the divisor to write the final answer.
$\begin{align} Quotient + \frac{Remainder}{Divisor} &= x + 1 + \frac{6}{x-1} \end{align}$

Compare the final answer with the given options.
The correct answer is option c, which is: $(x+1)+\frac{6}{x-1}$