To solve the problem [tex]\(\left(x^2 + 3x + 30\right) \div (x + 3)\)[/tex], we can use polynomial long division. Here's a step-by-step approach:
1. Set up the division:
The polynomial we are dividing is [tex]\(x^2 + 3x + 30\)[/tex] (the dividend), and we are dividing by [tex]\(x + 3\)[/tex] (the divisor).
2. First division:
- Divide the leading term of the dividend by the leading term of the divisor. [tex]\(\frac{x^2}{x} = x\)[/tex].
- Multiply [tex]\(x\)[/tex] by the entire divisor: [tex]\(x \cdot (x + 3) = x^2 + 3x\)[/tex].
- Subtract this from the original polynomial: [tex]\((x^2 + 3x + 30) - (x^2 + 3x) = 30\)[/tex].
3. Second division:
- The next term in the new polynomial is [tex]\(30\)[/tex].
- Divide the leading term of the new dividend (which is [tex]\(30\)[/tex]) by the leading term of the divisor ([tex]\(x\)[/tex]): [tex]\(\frac{30}{x}\)[/tex] is not possible since [tex]\(30\)[/tex] has no [tex]\(x\)[/tex] term to divide.
As a result, we can't perform a further division. Therefore, the remainder is [tex]\(30\)[/tex] as it cannot be divided by [tex]\(x + 3\)[/tex].
4. Conclusion:
The quotient of the division is [tex]\(x\)[/tex], and the remainder is [tex]\(30\)[/tex].
Putting it all together,
[tex]\[
\frac{x^2 + 3x + 30}{x + 3} = x + \frac{30}{x + 3}
\][/tex]
But in polynomial division terms, this would be:
[tex]\[
\left(x^2 + 3x + 30\right) \div (x + 3) = x \text{ with a remainder of } 30
\][/tex]
or more formally expressed:
[tex]\[
x^2 + 3x + 30 = (x + 3)(x) + 30
\][/tex]
