Math / Algebra

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Given that $f(x)=x^{2}+14 x+45$ and $g(x)=x+9$ , find $f(x) \div g(x)$ and express the result as a polynomial in simplest form.

Studdy Solution

STEP 1

1. We are given two polynomials: $f(x) = x^2 + 14x + 45$ and $g(x) = x + 9$ .
2. We need to perform polynomial division to find $\frac{f(x)}{g(x)}$ .
3. The result should be expressed as a polynomial in simplest form.

STEP 2

1. Set up the polynomial division $\frac{f(x)}{g(x)}$ .
2. Perform the polynomial division step-by-step.
3. Simplify the result to obtain the quotient polynomial.

STEP 3

Set up the polynomial division:
$\frac{x^2 + 14x + 45}{x + 9}$

STEP 4

Determine how many times the leading term of $g(x)$ divides into the leading term of $f(x)$ :
$\frac{x^2}{x} = x$

STEP 5

Multiply $x$ by $g(x)$ and subtract from $f(x)$ :
$x \cdot (x + 9) = x^2 + 9x$ $(x^2 + 14x + 45) - (x^2 + 9x) = 5x + 45$

STEP 6

Repeat the process with the new polynomial $5x + 45$ :
$\frac{5x}{x} = 5$

STEP 7

Multiply $5$ by $g(x)$ and subtract from the remaining polynomial:
$5 \cdot (x + 9) = 5x + 45$ $(5x + 45) - (5x + 45) = 0$

STEP 8

Combine the results of the division:
$\frac{x^2 + 14x + 45}{x + 9} = x + 5$
Solution: $\frac{f(x)}{g(x)} = x + 5$

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QuestionQuestion Watch Video Show Examples
Given that $f(x)=x^{2}+14 x+45$ and $g(x)=x+9$ , find $f(x) \div g(x)$ and express the result as a polynomial in simplest form.

Studdy Solution

STEP 1

1. We are given two polynomials: $f(x) = x^2 + 14x + 45$ and $g(x) = x + 9$ .
2. We need to perform polynomial division to find $\frac{f(x)}{g(x)}$ .
3. The result should be expressed as a polynomial in simplest form.

STEP 2

1. Set up the polynomial division $\frac{f(x)}{g(x)}$ .
2. Perform the polynomial division step-by-step.
3. Simplify the result to obtain the quotient polynomial.

STEP 3

Set up the polynomial division:
$\frac{x^2 + 14x + 45}{x + 9}$

STEP 4

Determine how many times the leading term of $g(x)$ divides into the leading term of $f(x)$ :
$\frac{x^2}{x} = x$

STEP 5

Multiply $x$ by $g(x)$ and subtract from $f(x)$ :
$x \cdot (x + 9) = x^2 + 9x$ $(x^2 + 14x + 45) - (x^2 + 9x) = 5x + 45$

STEP 6

Repeat the process with the new polynomial $5x + 45$ :
$\frac{5x}{x} = 5$

STEP 7

Multiply $5$ by $g(x)$ and subtract from the remaining polynomial:
$5 \cdot (x + 9) = 5x + 45$ $(5x + 45) - (5x + 45) = 0$

STEP 8

Combine the results of the division:
$\frac{x^2 + 14x + 45}{x + 9} = x + 5$
Solution: $\frac{f(x)}{g(x)} = x + 5$

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QuestionQuestion Watch Video Show ExamplesGiven that f(x)=x2+14x+45f(x)=x^{2}+14 x+45f(x)=x2+14x+45 and g(x)=x+9g(x)=x+9g(x)=x+9, find f(x)÷g(x)f(x) \div g(x)f(x)÷g(x) and express the result as a polynomial in simplest form.

Studdy Solution

STEP 1

1. We are given two polynomials: f(x)=x2+14x+45 f(x) = x^2 + 14x + 45 f(x)=x2+14x+45 and g(x)=x+9 g(x) = x + 9 g(x)=x+9.2. We need to perform polynomial division to find f(x)g(x) \frac{f(x)}{g(x)} g(x)f(x)​.3. The result should be expressed as a polynomial in simplest form.

STEP 2

1. Set up the polynomial division f(x)g(x) \frac{f(x)}{g(x)} g(x)f(x)​.2. Perform the polynomial division step-by-step.3. Simplify the result to obtain the quotient polynomial.

STEP 3

Set up the polynomial division:x2+14x+45x+9 \frac{x^2 + 14x + 45}{x + 9} x+9x2+14x+45​

STEP 4

Determine how many times the leading term of g(x) g(x) g(x) divides into the leading term of f(x) f(x) f(x):x2x=x \frac{x^2}{x} = x xx2​=x

STEP 5

Multiply x x x by g(x) g(x) g(x) and subtract from f(x) f(x) f(x):x⋅(x+9)=x2+9x x \cdot (x + 9) = x^2 + 9x x⋅(x+9)=x2+9x (x2+14x+45)−(x2+9x)=5x+45 (x^2 + 14x + 45) - (x^2 + 9x) = 5x + 45 (x2+14x+45)−(x2+9x)=5x+45

STEP 6

Repeat the process with the new polynomial 5x+45 5x + 45 5x+45:5xx=5 \frac{5x}{x} = 5 x5x​=5

STEP 7

Multiply 5 5 5 by g(x) g(x) g(x) and subtract from the remaining polynomial:5⋅(x+9)=5x+45 5 \cdot (x + 9) = 5x + 45 5⋅(x+9)=5x+45 (5x+45)−(5x+45)=0 (5x + 45) - (5x + 45) = 0 (5x+45)−(5x+45)=0

STEP 8

Combine the results of the division:x2+14x+45x+9=x+5 \frac{x^2 + 14x + 45}{x + 9} = x + 5 x+9x2+14x+45​=x+5Solution: f(x)g(x)=x+5 \frac{f(x)}{g(x)} = x + 5 g(x)f(x)​=x+5

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QuestionQuestion Watch Video Show Examples
Given that $f(x)=x^{2}+14 x+45$ and $g(x)=x+9$ , find $f(x) \div g(x)$ and express the result as a polynomial in simplest form.

1. We are given two polynomials: $f(x) = x^2 + 14x + 45$ and $g(x) = x + 9$ .
2. We need to perform polynomial division to find $\frac{f(x)}{g(x)}$ .
3. The result should be expressed as a polynomial in simplest form.

1. Set up the polynomial division $\frac{f(x)}{g(x)}$ .
2. Perform the polynomial division step-by-step.
3. Simplify the result to obtain the quotient polynomial.

Set up the polynomial division:
$\frac{x^2 + 14x + 45}{x + 9}$

Determine how many times the leading term of $g(x)$ divides into the leading term of $f(x)$ :
$\frac{x^2}{x} = x$

Multiply $x$ by $g(x)$ and subtract from $f(x)$ :
$x \cdot (x + 9) = x^2 + 9x$ $(x^2 + 14x + 45) - (x^2 + 9x) = 5x + 45$

Repeat the process with the new polynomial $5x + 45$ :
$\frac{5x}{x} = 5$

Multiply $5$ by $g(x)$ and subtract from the remaining polynomial:
$5 \cdot (x + 9) = 5x + 45$ $(5x + 45) - (5x + 45) = 0$

Combine the results of the division:
$\frac{x^2 + 14x + 45}{x + 9} = x + 5$
Solution: $\frac{f(x)}{g(x)} = x + 5$