To solve the polynomial division [tex]\((9x^3 + 24x^2 + 19x + 11) \div (3x + 4)\)[/tex], we follow these steps:
1. Set up the division:
- Dividend: [tex]\(9x^3 + 24x^2 + 19x + 11\)[/tex]
- Divisor: [tex]\(3x + 4\)[/tex]
2. Divide the leading term of the dividend by the leading term of the divisor:
- Leading term of the dividend: [tex]\(9x^3\)[/tex]
- Leading term of the divisor: [tex]\(3x\)[/tex]
- Quotient leading term: [tex]\(\frac{9x^3}{3x} = 3x^2\)[/tex]
3. Multiply the entire divisor by this quotient:
- [tex]\((3x + 4) \cdot (3x^2) = 9x^3 + 12x^2\)[/tex]
4. Subtract the result from the original polynomial:
[tex]\[
(9x^3 + 24x^2 + 19x + 11) - (9x^3 + 12x^2) = 12x^2 + 19x + 11
\][/tex]
5. Repeat the process with the new polynomial [tex]\(12x^2 + 19x + 11\)[/tex]:
- Leading term of the new polynomial: [tex]\(12x^2\)[/tex]
- Leading term of the divisor: [tex]\(3x\)[/tex]
- Quotient term: [tex]\(\frac{12x^2}{3x} = 4x\)[/tex]
6. Multiply the entire divisor by this new quotient term:
- [tex]\((3x + 4) \cdot 4x = 12x^2 + 16x\)[/tex]
7. Subtract the result from the new polynomial:
[tex]\[
(12x^2 + 19x + 11) - (12x^2 + 16x) = 3x + 11
\][/tex]
8. Repeat the process:
- Leading term of the new polynomial: [tex]\(3x\)[/tex]
- Leading term of the divisor: [tex]\(3x\)[/tex]
- Quotient term: [tex]\(\frac{3x}{3x} = 1\)[/tex]
9. Multiply the entire divisor by this quotient term:
- [tex]\((3x + 4) \cdot 1 = 3x + 4\)[/tex]
10. Subtract the result from the new polynomial:
[tex]\[
(3x + 11) - (3x + 4) = 7
\][/tex]
After performing these steps, we find:
- The quotient is [tex]\(3x^2 + 4x + 1\)[/tex]
- The remainder is [tex]\(7\)[/tex]
Therefore, the quotient and remainder of the division are:
[tex]\[
\text{Quotient: } 3x^2 + 4x + 1
\][/tex]
[tex]\[
\text{Remainder: } 7
\][/tex]
So, the final answer is:
[tex]\[
\left( 9x^3 + 24x^2 + 19x + 11 \right) \div \left( 3x + 4 \right) = \left( 3x^2 + 4x + 1, \text{ remainder: } 7 \right)
\][/tex]
