Answer :
Certainly! Let's go through the step-by-step process of dividing [tex]\(6x^2 + 23x + 18\)[/tex] by [tex]\(x + 3\)[/tex] using polynomial long division.
### Step-by-step Polynomial Long Division:
1. Setup the Division:
Set up the division problem like a long division problem with [tex]\(6x^2 + 23x + 18\)[/tex] (dividend) inside the division symbol and [tex]\(x + 3\)[/tex] (divisor) outside.
[tex]\[ \begin{array}{r|lr} x + 3 & 6x^2 + 23x + 18 \\ \end{array} \][/tex]
2. Divide the leading terms:
Divide the leading term of the dividend [tex]\(6x^2\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]:
[tex]\[ \frac{6x^2}{x} = 6x \][/tex]
Write [tex]\(6x\)[/tex] on top of the division symbol.
[tex]\[ \begin{array}{r|lr} x + 3 & 6x^2 + 23x + 18 \\ \hline & 6x \\ \end{array} \][/tex]
3. Multiply and Subtract:
Multiply [tex]\(6x\)[/tex] by [tex]\(x + 3\)[/tex]:
[tex]\[ 6x \cdot (x + 3) = 6x^2 + 18x \][/tex]
Subtract this from the original dividend:
[tex]\[ 6x^2 + 23x + 18 -(6x^2 + 18x) = 5x + 18 \][/tex]
Write the result below:
[tex]\[ \begin{array}{r|lr} x + 3 & 6x^2 + 23x + 18 \\ \hline & 6x \\ - (6x^2 + 18x) & 5x + 18 \\ \end{array} \][/tex]
4. Repeat the Process:
Divide the new leading term [tex]\(5x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[ \frac{5x}{x} = 5 \][/tex]
Write [tex]\(5\)[/tex] on top next to [tex]\(6x\)[/tex]:
[tex]\[ \begin{array}{r|lr} x + 3 & 6x^2 + 23x + 18 \\ \hline & 6x + 5 \\ \end{array} \][/tex]
Multiply [tex]\(5\)[/tex] by [tex]\(x + 3\)[/tex]:
[tex]\[ 5 \cdot (x + 3) = 5x + 15 \][/tex]
Subtract this from the current remainder:
[tex]\[ 5x + 18 - (5x + 15) = 3 \][/tex]
Write the result below:
[tex]\[ \begin{array}{r|lr} x + 3 & 6x^2 + 23x + 18 \\ \hline & 6x + 5 \\ - (6x^2 + 18x) & 5x + 18 \\ - (5x + 15) & 3 \\ \end{array} \][/tex]
5. Final Quotient and Remainder:
We have no more terms to bring down. So, the polynomial division is complete, and we have:
- Quotient: [tex]\(6x + 5\)[/tex]
- Remainder: [tex]\(3\)[/tex]
### Conclusion:
The result of dividing [tex]\(6x^2 + 23x + 18\)[/tex] by [tex]\(x + 3\)[/tex] produces a quotient of [tex]\(6x + 5\)[/tex] and a remainder of [tex]\(3\)[/tex]. Therefore, we can express the result as:
[tex]\[ 6x^2 + 23x + 18 = (x + 3)(6x + 5) + 3 \][/tex]
So the quotient and remainder are:
[tex]\[ \boxed{6x + 5 \text{ (quotient)}, \; 3 \text{ (remainder)}} \][/tex]
### Step-by-step Polynomial Long Division:
1. Setup the Division:
Set up the division problem like a long division problem with [tex]\(6x^2 + 23x + 18\)[/tex] (dividend) inside the division symbol and [tex]\(x + 3\)[/tex] (divisor) outside.
[tex]\[ \begin{array}{r|lr} x + 3 & 6x^2 + 23x + 18 \\ \end{array} \][/tex]
2. Divide the leading terms:
Divide the leading term of the dividend [tex]\(6x^2\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]:
[tex]\[ \frac{6x^2}{x} = 6x \][/tex]
Write [tex]\(6x\)[/tex] on top of the division symbol.
[tex]\[ \begin{array}{r|lr} x + 3 & 6x^2 + 23x + 18 \\ \hline & 6x \\ \end{array} \][/tex]
3. Multiply and Subtract:
Multiply [tex]\(6x\)[/tex] by [tex]\(x + 3\)[/tex]:
[tex]\[ 6x \cdot (x + 3) = 6x^2 + 18x \][/tex]
Subtract this from the original dividend:
[tex]\[ 6x^2 + 23x + 18 -(6x^2 + 18x) = 5x + 18 \][/tex]
Write the result below:
[tex]\[ \begin{array}{r|lr} x + 3 & 6x^2 + 23x + 18 \\ \hline & 6x \\ - (6x^2 + 18x) & 5x + 18 \\ \end{array} \][/tex]
4. Repeat the Process:
Divide the new leading term [tex]\(5x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[ \frac{5x}{x} = 5 \][/tex]
Write [tex]\(5\)[/tex] on top next to [tex]\(6x\)[/tex]:
[tex]\[ \begin{array}{r|lr} x + 3 & 6x^2 + 23x + 18 \\ \hline & 6x + 5 \\ \end{array} \][/tex]
Multiply [tex]\(5\)[/tex] by [tex]\(x + 3\)[/tex]:
[tex]\[ 5 \cdot (x + 3) = 5x + 15 \][/tex]
Subtract this from the current remainder:
[tex]\[ 5x + 18 - (5x + 15) = 3 \][/tex]
Write the result below:
[tex]\[ \begin{array}{r|lr} x + 3 & 6x^2 + 23x + 18 \\ \hline & 6x + 5 \\ - (6x^2 + 18x) & 5x + 18 \\ - (5x + 15) & 3 \\ \end{array} \][/tex]
5. Final Quotient and Remainder:
We have no more terms to bring down. So, the polynomial division is complete, and we have:
- Quotient: [tex]\(6x + 5\)[/tex]
- Remainder: [tex]\(3\)[/tex]
### Conclusion:
The result of dividing [tex]\(6x^2 + 23x + 18\)[/tex] by [tex]\(x + 3\)[/tex] produces a quotient of [tex]\(6x + 5\)[/tex] and a remainder of [tex]\(3\)[/tex]. Therefore, we can express the result as:
[tex]\[ 6x^2 + 23x + 18 = (x + 3)(6x + 5) + 3 \][/tex]
So the quotient and remainder are:
[tex]\[ \boxed{6x + 5 \text{ (quotient)}, \; 3 \text{ (remainder)}} \][/tex]
