Certainly! Let's solve the division of the polynomial [tex]\( \left(x^2 + 3x + 30\right) \div (x + 3) \)[/tex] using polynomial long division.
### Step-by-Step Solution:
1. Set up the division: We divide the polynomial [tex]\( x^2 + 3x + 30 \)[/tex] by [tex]\( x + 3 \)[/tex].
``` ________ x + 3 | x^2 + 3x + 30 ```
2. First Division: - Divide the leading term of the numerator ([tex]\(x^2\)[/tex]) by the leading term of the denominator ([tex]\(x\)[/tex]): [tex]\[
\frac{x^2}{x} = x
\][/tex] - The quotient so far is [tex]\( x \)[/tex].
``` x ________ x + 3 | x^2 + 3x + 30 ```
3. Multiply and Subtract: - Multiply [tex]\(x\)[/tex] by [tex]\(x + 3\)[/tex]: [tex]\[
x \cdot (x + 3) = x^2 + 3x
\][/tex] - Subtract this product from the original polynomial: [tex]\[
(x^2 + 3x + 30) - (x^2 + 3x) = 30
\][/tex]
4. Next Division: - Since there is no more [tex]\(x\)[/tex] term in the polynomial 30, we finish the division here. - The remainder is [tex]\(30\)[/tex].
### Conclusion:
- The quotient is [tex]\( x \)[/tex]. - The remainder is [tex]\( 30 \)[/tex].
