To solve the division of the polynomial expression [tex]\(\frac{3x^3 + 10x^2 + 7x}{x + 1}\)[/tex], we need to perform polynomial division. Let's go through the steps in detail:
1. Set up the division: We are dividing [tex]\(3x^3 + 10x^2 + 7x\)[/tex] by [tex]\(x + 1\)[/tex].
2. Divide the leading terms: The leading term of the numerator is [tex]\(3x^3\)[/tex], and the leading term of the denominator is [tex]\(x\)[/tex]. Dividing these gives:
[tex]\[
\frac{3x^3}{x} = 3x^2
\][/tex]
This is the first term of our quotient.
3. Multiply and subtract: Multiply the entire denominator [tex]\(x + 1\)[/tex] by the term [tex]\(3x^2\)[/tex] and subtract this product from the original polynomial:
[tex]\[
(3x^3 + 10x^2 + 7x) - (3x^2 \cdot (x + 1)) = 3x^3 + 10x^2 + 7x - (3x^3 + 3x^2) = (10x^2 - 3x^2) + 7x = 7x^2 + 7x
\][/tex]
4. Repeat the process: Now we work with the new polynomial [tex]\(7x^2 + 7x\)[/tex]. Divide the leading term [tex]\(7x^2\)[/tex] by the leading term of the divisor [tex]\(x\)[/tex]:
[tex]\[
\frac{7x^2}{x} = 7x
\][/tex]
This is the next term of our quotient.
5. Multiply and subtract again: Multiply the entire denominator [tex]\(x + 1\)[/tex] by [tex]\(7x\)[/tex] and subtract again:
[tex]\[
(7x^2 + 7x) - (7x \cdot (x + 1)) = 7x^2 + 7x - (7x^2 + 7x) = 0
\][/tex]
Since we are left with 0, there are no more terms to bring down, and the process stops here.
6. Quotient and remainder:
The quotient is [tex]\(3x^2 + 7x\)[/tex].
The remainder is [tex]\(0\)[/tex].
Thus, the final result of the division is given by the polynomial quotient [tex]\(3x^2 + 7x\)[/tex] with no remainder. Therefore, the expression simplifies to:
[tex]\[
\frac{3x^3 + 10x^2 + 7x}{x + 1} = 3x^2 + 7x
\][/tex]
So the final answer is:
[tex]\[
3x^2 + 7x
\][/tex]
