Question Rewrite the rational function $g(x) = \frac{x^2 + 7x - 12}{x + 2}$ in the form $g(x) = p(x) + \frac{r}{x + 2}$ , where $p(x)$ is a polynomial and $r$ is an integer.

Studdy Solution

STEP 1

Assumptions
1. The given function is $g(x)=\frac{x^{2}+7 x-12}{x+2}$
2. We need to rewrite the function in the form $g(x)=p(x)+\frac{r}{x+2}$ , where $p(x)$ is a polynomial and $r$ is an integer.

STEP 2

We can rewrite the function in the desired form by performing polynomial division on the numerator by the denominator.

STEP 3

Perform polynomial division of $x^{2}+7 x-12$ by $x+2$ .

STEP 4

The division can be set up as follows:
\begin{{array}}{{r|l}} x+2 & x^{2}+7x-12 \\ \end{{array}}

STEP 5

Divide the first term in the dividend ( $x^{2}$ ) by the first term in the divisor ( $x$ ) to get the first term of the quotient ( $x$ ).

STEP 6

Multiply the divisor ( $x+2$ ) by the first term of the quotient ( $x$ ) and subtract the result from the dividend.

STEP 7

The result of the subtraction is $3x-12$ .

STEP 8

Repeat the division process for the new dividend $3x-12$ .

STEP 9

Divide the first term in the new dividend ( $3x$ ) by the first term in the divisor ( $x$ ) to get the next term of the quotient ( $3$ ).

STEP 10

Multiply the divisor ( $x+2$ ) by the new term in the quotient ( $3$ ) and subtract the result from the new dividend.

STEP 11

The result of the subtraction is $-6$ .

STEP 12

Since the degree of the result (-6) is less than the degree of the divisor ( $x+2$ ), we stop the division process. The result (-6) is the remainder.

STEP 13

The quotient of the division is $x+3$ and the remainder is $-6$ .

STEP 14

So, the given function can be rewritten as $g(x)=p(x)+\frac{r}{x+2}$ , where $p(x)$ is the quotient and $r$ is the remainder.

STEP 15

Substitute the values of $p(x)$ and $r$ into the equation.
$g(x)=x+3-\frac{6}{x+2}$
So, the function $g(x)=\frac{x^{2}+7 x-12}{x+2}$ can be rewritten as $g(x)=x+3-\frac{6}{x+2}$ .

Was this helpful?

Question Rewrite the rational function $g(x) = \frac{x^2 + 7x - 12}{x + 2}$ in the form $g(x) = p(x) + \frac{r}{x + 2}$ , where $p(x)$ is a polynomial and $r$ is an integer.

Studdy Solution

STEP 1

Assumptions
1. The given function is $g(x)=\frac{x^{2}+7 x-12}{x+2}$
2. We need to rewrite the function in the form $g(x)=p(x)+\frac{r}{x+2}$ , where $p(x)$ is a polynomial and $r$ is an integer.

STEP 2

We can rewrite the function in the desired form by performing polynomial division on the numerator by the denominator.

STEP 3

Perform polynomial division of $x^{2}+7 x-12$ by $x+2$ .

STEP 4

The division can be set up as follows:
\begin{{array}}{{r|l}} x+2 & x^{2}+7x-12 \\ \end{{array}}

STEP 5

Divide the first term in the dividend ( $x^{2}$ ) by the first term in the divisor ( $x$ ) to get the first term of the quotient ( $x$ ).

STEP 6

Multiply the divisor ( $x+2$ ) by the first term of the quotient ( $x$ ) and subtract the result from the dividend.

STEP 7

The result of the subtraction is $3x-12$ .

STEP 8

Repeat the division process for the new dividend $3x-12$ .

STEP 9

Divide the first term in the new dividend ( $3x$ ) by the first term in the divisor ( $x$ ) to get the next term of the quotient ( $3$ ).

STEP 10

Multiply the divisor ( $x+2$ ) by the new term in the quotient ( $3$ ) and subtract the result from the new dividend.

STEP 11

The result of the subtraction is $-6$ .

STEP 12

Since the degree of the result (-6) is less than the degree of the divisor ( $x+2$ ), we stop the division process. The result (-6) is the remainder.

STEP 13

The quotient of the division is $x+3$ and the remainder is $-6$ .

STEP 14

So, the given function can be rewritten as $g(x)=p(x)+\frac{r}{x+2}$ , where $p(x)$ is the quotient and $r$ is the remainder.

STEP 15

Substitute the values of $p(x)$ and $r$ into the equation.
$g(x)=x+3-\frac{6}{x+2}$
So, the function $g(x)=\frac{x^{2}+7 x-12}{x+2}$ can be rewritten as $g(x)=x+3-\frac{6}{x+2}$ .

Get Studdy

Studdy solves anything!

Start learning now

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

Question Rewrite the rational function g(x)=x2+7x−12x+2g(x) = \frac{x^2 + 7x - 12}{x + 2}g(x)=x+2x2+7x−12​ in the form g(x)=p(x)+rx+2g(x) = p(x) + \frac{r}{x + 2}g(x)=p(x)+x+2r​, where p(x)p(x)p(x) is a polynomial and rrr is an integer.

Studdy Solution

STEP 1

Assumptions1. The given function is g(x)=x2+7x−12x+2g(x)=\frac{x^{2}+7 x-12}{x+2}g(x)=x+2x2+7x−12​2. We need to rewrite the function in the form g(x)=p(x)+rx+2g(x)=p(x)+\frac{r}{x+2}g(x)=p(x)+x+2r​, where p(x)p(x)p(x) is a polynomial and rrr is an integer.

STEP 2

We can rewrite the function in the desired form by performing polynomial division on the numerator by the denominator.

STEP 3

Perform polynomial division of x2+7x−12x^{2}+7 x-12x2+7x−12 by x+2x+2x+2.

STEP 4

The division can be set up as follows:\begin{{array}}{{r|l}} x+2 & x^{2}+7x-12 \\ \end{{array}}

STEP 5

Divide the first term in the dividend (x2x^{2}x2) by the first term in the divisor (xxx) to get the first term of the quotient (xxx).

STEP 6

Multiply the divisor (x+2x+2x+2) by the first term of the quotient (xxx) and subtract the result from the dividend.

STEP 7

The result of the subtraction is 3x−123x-123x−12.

STEP 8

Repeat the division process for the new dividend 3x−123x-123x−12.

STEP 9

Divide the first term in the new dividend (3x3x3x) by the first term in the divisor (xxx) to get the next term of the quotient (333).

STEP 10

Multiply the divisor (x+2x+2x+2) by the new term in the quotient (333) and subtract the result from the new dividend.

STEP 11

The result of the subtraction is −6-6−6.

STEP 12

Since the degree of the result (-6) is less than the degree of the divisor (x+2x+2x+2), we stop the division process. The result (-6) is the remainder.

STEP 13

The quotient of the division is x+3x+3x+3 and the remainder is −6-6−6.

STEP 14

So, the given function can be rewritten as g(x)=p(x)+rx+2g(x)=p(x)+\frac{r}{x+2}g(x)=p(x)+x+2r​, where p(x)p(x)p(x) is the quotient and rrr is the remainder.

STEP 15

Substitute the values of p(x)p(x)p(x) and rrr into the equation.g(x)=x+3−6x+2g(x)=x+3-\frac{6}{x+2}g(x)=x+3−x+26​So, the function g(x)=x2+7x−12x+2g(x)=\frac{x^{2}+7 x-12}{x+2}g(x)=x+2x2+7x−12​ can be rewritten as g(x)=x+3−6x+2g(x)=x+3-\frac{6}{x+2}g(x)=x+3−x+26​.

Get Studdy

Studdy solves anything!

Start learning now

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

Question Rewrite the rational function $g(x) = \frac{x^2 + 7x - 12}{x + 2}$ in the form $g(x) = p(x) + \frac{r}{x + 2}$ , where $p(x)$ is a polynomial and $r$ is an integer.

Assumptions
1. The given function is $g(x)=\frac{x^{2}+7 x-12}{x+2}$
2. We need to rewrite the function in the form $g(x)=p(x)+\frac{r}{x+2}$ , where $p(x)$ is a polynomial and $r$ is an integer.

Perform polynomial division of $x^{2}+7 x-12$ by $x+2$ .

The division can be set up as follows:
\begin{{array}}{{r|l}} x+2 & x^{2}+7x-12 \\ \end{{array}}

Divide the first term in the dividend ( $x^{2}$ ) by the first term in the divisor ( $x$ ) to get the first term of the quotient ( $x$ ).

Multiply the divisor ( $x+2$ ) by the first term of the quotient ( $x$ ) and subtract the result from the dividend.

The result of the subtraction is $3x-12$ .

Repeat the division process for the new dividend $3x-12$ .

Divide the first term in the new dividend ( $3x$ ) by the first term in the divisor ( $x$ ) to get the next term of the quotient ( $3$ ).

Multiply the divisor ( $x+2$ ) by the new term in the quotient ( $3$ ) and subtract the result from the new dividend.

The result of the subtraction is $-6$ .

Since the degree of the result (-6) is less than the degree of the divisor ( $x+2$ ), we stop the division process. The result (-6) is the remainder.

The quotient of the division is $x+3$ and the remainder is $-6$ .

So, the given function can be rewritten as $g(x)=p(x)+\frac{r}{x+2}$ , where $p(x)$ is the quotient and $r$ is the remainder.

Substitute the values of $p(x)$ and $r$ into the equation.
$g(x)=x+3-\frac{6}{x+2}$
So, the function $g(x)=\frac{x^{2}+7 x-12}{x+2}$ can be rewritten as $g(x)=x+3-\frac{6}{x+2}$ .