Sure! Let's perform the polynomial long division step-by-step to divide [tex]\(3x^2 + 9x + 4\)[/tex] by [tex]\(x + 2\)[/tex].
1. Setup the Division:
We are dividing [tex]\(3x^2 + 9x + 4\)[/tex] by [tex]\(x + 2\)[/tex].
2. First Division Step:
- Divide the first term of the dividend by the first term of the divisor: [tex]\(\frac{3x^2}{x} = 3x\)[/tex].
- So, the first term of the quotient is [tex]\(3x\)[/tex].
3. First Subtraction Step:
- Multiply [tex]\(3x\)[/tex] by the entire divisor [tex]\(x + 2\)[/tex]: [tex]\(3x \cdot (x + 2) = 3x^2 + 6x\)[/tex].
- Subtract this from the original dividend: [tex]\((3x^2 + 9x + 4) - (3x^2 + 6x)\)[/tex].
- This results in: [tex]\((3x^2 + 9x + 4) - (3x^2 + 6x) = 3x\)[/tex].
4. Second Division Step:
- Divide the first term of the new polynomial by the first term of the divisor: [tex]\(\frac{3x}{x} = 3\)[/tex].
- So, the second term of the quotient is [tex]\(3\)[/tex].
5. Second Subtraction Step:
- Multiply [tex]\(3\)[/tex] by the entire divisor [tex]\(x + 2\)[/tex]: [tex]\(3 \cdot (x + 2) = 3x + 6\)[/tex].
- Subtract this from the new polynomial: [tex]\((3x + 4) - (3x + 6)\)[/tex].
- This results in: [tex]\((3x + 4) - (3x + 6) = -2\)[/tex].
6. Conclusion:
- Now there are no more terms left in the dividend to bring down, and the degree of the remainder [tex]\(-2\)[/tex] is less than the degree of the divisor [tex]\(x + 2\)[/tex].
- Thus, the polynomial long division process is complete.
The quotient is:
[tex]\[
3x + 3
\][/tex]
And the remainder is:
[tex]\[
-2
\][/tex]
So, the quotient of the division [tex]\( \left(3x^2 + 9x + 4\right) \div (x + 2) \)[/tex] is [tex]\( 3x + 3 \)[/tex] and the remainder is [tex]\( -2 \)[/tex].
