To find the quotient and remainder of the polynomial division [tex]\((3x^2 + 8x - 3) \div (x + 3)\)[/tex], follow these steps:
1. Setup the Division:
We want to divide the polynomial [tex]\(3x^2 + 8x - 3\)[/tex] by [tex]\(x + 3\)[/tex].
2. Divide the First Terms:
Divide the leading term of the dividend [tex]\(3x^2\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]:
[tex]\[\frac{3x^2}{x} = 3x\][/tex]
This is the first term of the quotient.
3. Multiply and Subtract:
Multiply the entire divisor [tex]\(x + 3\)[/tex] by the first term of the quotient [tex]\(3x\)[/tex]:
[tex]\[3x \cdot (x + 3) = 3x^2 + 9x\][/tex]
Subtract this result from the original polynomial:
[tex]\[
(3x^2 + 8x - 3) - (3x^2 + 9x) = 8x - 9x - 3 = -x - 3
\][/tex]
4. Repeat the Process:
Now take the next term [tex]\(-x - 3\)[/tex] and divide the leading term [tex]\(-x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[\frac{-x}{x} = -1\][/tex]
This becomes the next term of the quotient.
5. Multiply and Subtract Again:
Multiply the entire divisor [tex]\(x + 3\)[/tex] by [tex]\(-1\)[/tex]:
[tex]\[-1 \cdot (x + 3) = -x - 3\][/tex]
Subtract this from the current result:
[tex]\[
(-x - 3) - (-x - 3) = 0
\][/tex]
Since there is no remainder, our quotient is complete.
The quotient is:
[tex]\[3x - 1\][/tex]
Thus, the quotient of [tex]\((3x^2 + 8x - 3) \div (x + 3)\)[/tex] is [tex]\(\boxed{3x - 1}\)[/tex].
