Use long division to divide. Express any numbers as integers or simplified fractions.

Ask by Harper Ortega. in the United States

Mar 22,2025

Question

Use long division to divide. Express any numbers as integers or simplified fractions. $$ \left(-6+3 x+27 x^{2}+22 x^{3}+5 x^{4}\right) \div(x+2)=\square $$

UpStudy AI · Accepted Answer

1. Write the dividend in standard form:
$$
5x^4 + 22x^3 + 27x^2 + 3x - 6
$$
We are dividing by:
$$
x+2
$$

2. Set up the long division:

$$
\require{enclose}
\begin{array}{rll}
5x^3+12x^2+3x-3 & \enclose{longdiv}{5x^4+22x^3+27x^2+3x-6} \
\end{array}
$$

3. Divide the first term of the dividend by the first term of the divisor:
$$
\frac{5x^4}{x} = 5x^3.
$$
Write $$5x^3$$ above the long division bar.

4. Multiply $$5x^3$$ by $$x + 2$$:
$$
5x^3 \cdot (x+2) = 5x^4 + 10x^3.
$$

5. Subtract this from the dividend:
$$
\begin{array}{rcl}
(5x^4 + 22x^3) - (5x^4 + 10x^3) & = & 12x^3.
\end{array}
$$
Bring down the next term $$+27x^2$$ so the new expression is:
$$
12x^3 + 27x^2.
$$

6. Divide the leading term of the new expression by $$x$$:
$$
\frac{12x^3}{x} = 12x^2.
$$
Write $$12x^2$$ above the division bar next to $$5x^3$$.

7. Multiply $$12x^2$$ by $$x + 2$$:
$$
12x^2 \cdot (x+2) = 12x^3 + 24x^2.
$$

8. Subtract:
$$
\begin{array}{rcl}
(12x^3 + 27x^2) - (12x^3 + 24x^2) & = & 3x^2.
\end{array}
$$
Bring down the next term $$+3x$$ resulting in:
$$
3x^2 + 3x.
$$

9. Divide the leading term:
$$
\frac{3x^2}{x} = 3x.
$$
Write $$3x$$ above the bar.

10. Multiply $$3x$$ by $$x+2$$:
$$
3x \cdot (x+2) = 3x^2+6x.
$$

11. Subtract:
$$
\begin{array}{rcl}
(3x^2+3x) - (3x^2+6x) & = & -3x.
\end{array}
$$
Bring down the next term $$-6$$ to get:
$$
-3x - 6.
$$

12. Divide:
$$
\frac{-3x}{x} = -3.
$$
Write $$-3$$ above the bar.

13. Multiply $$-3$$ by $$x+2$$:
$$
-3 \cdot (x+2) = -3x - 6.
$$

14. Subtract:
$$
(-3x-6) - (-3x-6) = 0.
$$
The remainder is $$0$$.

Thus, the quotient is:
$$
5x^3+12x^2+3x-3.
$$

The final result is:
$$
\boxed{5x^3+12x^2+3x-3}.
$$
The result of dividing $$ -6 + 3x + 27x^2 + 22x^3 + 5x^4 $$ by $$ x + 2 $$ is $$ 5x^3 + 12x^2 + 3x - 3 $$.

Use long division to divide. Express any numbers as integers or simplified fractions.

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