Math / Algebra

QuestionQuestion Watch Video Show Examples Given that $f(x)=x^{2}+7 x-18$ and $g(x)=x+9$ , find $(f \div g)(x)$ and express the result as a polynomial in simplest form.

Studdy Solution

STEP 1

1. The function $f(x) = x^2 + 7x - 18$ is a polynomial of degree 2.
2. The function $g(x) = x + 9$ is a polynomial of degree 1.
3. We need to perform polynomial division to find $(f \div g)(x)$ and express the result in simplest form.

STEP 2

1. Set up the polynomial division of $f(x)$ by $g(x)$ .
2. Perform the polynomial division step-by-step.
3. Simplify the result and express it as a polynomial.

STEP 3

Set up the polynomial division of $f(x)$ by $g(x)$ .
We are dividing $x^2 + 7x - 18$ by $x + 9$ .
$\frac{x^2 + 7x - 18}{x + 9}$

STEP 4

Determine the first term of the quotient by dividing the leading term of the numerator by the leading term of the denominator.
$\frac{x^2}{x} = x$

STEP 5

Multiply $x$ by the divisor $g(x)$ and subtract the result from $f(x)$ .
$x^2 + 7x - 18 - (x(x + 9)) = x^2 + 7x - 18 - (x^2 + 9x)$ $x^2 + 7x - 18 - x^2 - 9x = -2x - 18$

STEP 6

Determine the next term of the quotient by dividing the new leading term of the remainder by the leading term of the divisor.
$\frac{-2x}{x} = -2$

STEP 7

Multiply $-2$ by the divisor $g(x)$ and subtract the result from the current remainder.
$-2x - 18 - (-2(x + 9)) = -2x - 18 - (-2x - 18)$ $-2x - 18 + 2x + 18 = 0$

STEP 8

Combine the terms of the quotient to get the result of the division.
The quotient is:
$x - 2$
Solution: The quotient of the division $(f \div g)(x)$ is $x - 2$ .

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QuestionQuestion Watch Video Show Examples Given that $f(x)=x^{2}+7 x-18$ and $g(x)=x+9$ , find $(f \div g)(x)$ and express the result as a polynomial in simplest form.

Studdy Solution

STEP 1

1. The function $f(x) = x^2 + 7x - 18$ is a polynomial of degree 2.
2. The function $g(x) = x + 9$ is a polynomial of degree 1.
3. We need to perform polynomial division to find $(f \div g)(x)$ and express the result in simplest form.

STEP 2

1. Set up the polynomial division of $f(x)$ by $g(x)$ .
2. Perform the polynomial division step-by-step.
3. Simplify the result and express it as a polynomial.

STEP 3

Set up the polynomial division of $f(x)$ by $g(x)$ .
We are dividing $x^2 + 7x - 18$ by $x + 9$ .
$\frac{x^2 + 7x - 18}{x + 9}$

STEP 4

Determine the first term of the quotient by dividing the leading term of the numerator by the leading term of the denominator.
$\frac{x^2}{x} = x$

STEP 5

Multiply $x$ by the divisor $g(x)$ and subtract the result from $f(x)$ .
$x^2 + 7x - 18 - (x(x + 9)) = x^2 + 7x - 18 - (x^2 + 9x)$ $x^2 + 7x - 18 - x^2 - 9x = -2x - 18$

STEP 6

Determine the next term of the quotient by dividing the new leading term of the remainder by the leading term of the divisor.
$\frac{-2x}{x} = -2$

STEP 7

Multiply $-2$ by the divisor $g(x)$ and subtract the result from the current remainder.
$-2x - 18 - (-2(x + 9)) = -2x - 18 - (-2x - 18)$ $-2x - 18 + 2x + 18 = 0$

STEP 8

Combine the terms of the quotient to get the result of the division.
The quotient is:
$x - 2$
Solution: The quotient of the division $(f \div g)(x)$ is $x - 2$ .

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QuestionQuestion Watch Video Show Examples Given that f(x)=x2+7x−18f(x)=x^{2}+7 x-18f(x)=x2+7x−18 and g(x)=x+9g(x)=x+9g(x)=x+9, find (f÷g)(x)(f \div g)(x)(f÷g)(x) and express the result as a polynomial in simplest form.

Studdy Solution

STEP 1

1. The function f(x)=x2+7x−18 f(x) = x^2 + 7x - 18 f(x)=x2+7x−18 is a polynomial of degree 2.2. The function g(x)=x+9 g(x) = x + 9 g(x)=x+9 is a polynomial of degree 1.3. We need to perform polynomial division to find (f÷g)(x) (f \div g)(x) (f÷g)(x) and express the result in simplest form.

STEP 2

1. Set up the polynomial division of f(x) f(x) f(x) by g(x) g(x) g(x).2. Perform the polynomial division step-by-step.3. Simplify the result and express it as a polynomial.

STEP 3

Set up the polynomial division of f(x) f(x) f(x) by g(x) g(x) g(x).We are dividing x2+7x−18 x^2 + 7x - 18 x2+7x−18 by x+9 x + 9 x+9.x2+7x−18x+9 \frac{x^2 + 7x - 18}{x + 9} x+9x2+7x−18​

STEP 4

Determine the first term of the quotient by dividing the leading term of the numerator by the leading term of the denominator.x2x=x \frac{x^2}{x} = x xx2​=x

STEP 5

STEP 6

Determine the next term of the quotient by dividing the new leading term of the remainder by the leading term of the divisor.−2xx=−2 \frac{-2x}{x} = -2 x−2x​=−2

STEP 7

STEP 8

Combine the terms of the quotient to get the result of the division.The quotient is:x−2 x - 2 x−2Solution: The quotient of the division (f÷g)(x) (f \div g)(x) (f÷g)(x) is x−2 x - 2 x−2.

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QuestionQuestion Watch Video Show Examples Given that $f(x)=x^{2}+7 x-18$ and $g(x)=x+9$ , find $(f \div g)(x)$ and express the result as a polynomial in simplest form.

1. The function $f(x) = x^2 + 7x - 18$ is a polynomial of degree 2.
2. The function $g(x) = x + 9$ is a polynomial of degree 1.
3. We need to perform polynomial division to find $(f \div g)(x)$ and express the result in simplest form.

1. Set up the polynomial division of $f(x)$ by $g(x)$ .
2. Perform the polynomial division step-by-step.
3. Simplify the result and express it as a polynomial.

Set up the polynomial division of $f(x)$ by $g(x)$ .
We are dividing $x^2 + 7x - 18$ by $x + 9$ .
$\frac{x^2 + 7x - 18}{x + 9}$

Determine the first term of the quotient by dividing the leading term of the numerator by the leading term of the denominator.
$\frac{x^2}{x} = x$

Determine the next term of the quotient by dividing the new leading term of the remainder by the leading term of the divisor.
$\frac{-2x}{x} = -2$

Combine the terms of the quotient to get the result of the division.
The quotient is:
$x - 2$
Solution: The quotient of the division $(f \div g)(x)$ is $x - 2$ .