Sure, let's solve the division step-by-step.
Given:
[tex]\[
\frac{12 x^2 + 21 x + 10}{3 x + 3}
\][/tex]
To divide polynomials, we perform polynomial long division. Here are the steps:
1. Divide the Leading Terms:
- The leading term of the numerator is [tex]\(12x^2\)[/tex] and the leading term of the denominator is [tex]\(3x\)[/tex].
- Divide [tex]\(12x^2\)[/tex] by [tex]\(3x\)[/tex] to get the first term of the quotient: [tex]\(\frac{12x^2}{3x} = 4x\)[/tex].
2. Multiply and Subtract:
- Multiply the entire denominator [tex]\(3x + 3\)[/tex] by [tex]\(4x\)[/tex]:
[tex]\[
(3x + 3) \cdot 4x = 12x^2 + 12x
\][/tex]
- Subtract this product from the original numerator:
[tex]\[
(12x^2 + 21x + 10) - (12x^2 + 12x) = (21x - 12x) + 10 = 9x + 10
\][/tex]
3. Repeat the Process:
- Now, the new numerator is [tex]\(9x + 10\)[/tex]. Divide the leading term [tex]\(9x\)[/tex] by the leading term of the denominator [tex]\(3x\)[/tex]:
[tex]\[
\frac{9x}{3x} = 3
\][/tex]
- Multiply the entire denominator [tex]\(3x + 3\)[/tex] by [tex]\(3\)[/tex]:
[tex]\[
(3x + 3) \cdot 3 = 9x + 9
\][/tex]
- Subtract this product from the new numerator:
[tex]\[
(9x + 10) - (9x + 9) = 10 - 9 = 1
\][/tex]
4. Conclusion:
- The quotient is the sum of the terms we found from dividing step-by-step: [tex]\(4x + 3\)[/tex].
- The remainder is the term left over, which is [tex]\(1\)[/tex].
Thus, the result of the division is:
[tex]\[
\frac{12 x^2 + 21 x + 10}{3 x + 3} = 4x + 3 \quad \text{with a remainder of} \quad 1
\][/tex]
