Divide. $$ \left(6 x^{3}+11 x^{2}+16 x+21\right) \div(3 x+4) $$ Your answer should give the quotient and the remainder: Quotient:

Question

UpStudy AI · Accepted Answer

Let us perform polynomial long division of
$$
6x^3 + 11x^2 + 16x + 21
$$
by
$$
3x+4.
$$

1. **First step:**
   - Divide the leading term $$6x^3$$ by $$3x$$:
     $$
     \frac{6x^3}{3x} = 2x^2.
     $$
   - Multiply $$2x^2$$ by the divisor:
     $$
     2x^2(3x+4) = 6x^3 + 8x^2.
     $$
   - Subtract this from the dividend:
     $$
     \begin{aligned}
     (6x^3 + 11x^2) - (6x^3 + 8x^2) &= 3x^2.
     \end{aligned}
     $$
   - Bring down the next term $$16x$$:
     $$
     3x^2 + 16x.
     $$

2. **Second step:**
   - Divide $$3x^2$$ by $$3x$$:
     $$
     \frac{3x^2}{3x} = x.
     $$
   - Multiply $$x$$ by the divisor:
     $$
     x(3x+4) = 3x^2 + 4x.
     $$
   - Subtract this from $$3x^2 + 16x$$:
     $$
     (3x^2 + 16x) - (3x^2 + 4x) = 12x.
     $$
   - Bring down the next term $$21$$:
     $$
     12x + 21.
     $$

3. **Third step:**
   - Divide $$12x$$ by $$3x$$:
     $$
     \frac{12x}{3x} = 4.
     $$
   - Multiply $$4$$ by the divisor:
     $$
     4(3x+4) = 12x + 16.
     $$
   - Subtract this from $$12x + 21$$:
     $$
     (12x + 21) - (12x + 16) = 5.
     $$

The division yields a quotient and a remainder:
- Quotient: $$2x^2 + x + 4$$
- Remainder: $$5$$

Thus, the result of the division is:
$$
\text{Quotient: } 2x^2 + x + 4, \quad \text{Remainder: } 5.
$$
Quotient: $$2x^2 + x + 4$$, Remainder: 5

Divide.

Your answer should give the quotient and the remainder:
Quotient:

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Divide. Your answer should give the quotient and the remainder: Quotient: