Question: Divide and check.
(20w² + 20w + 28w³ + 31) ÷ (4w + 4)
Write the answer as the quotient plus the remainder divided by the divisor.
(20w² + 20w + 28w³ + 31) ÷ (4w + 4) = □ (Simplify the answer.)
1. **State the problem:** Divide the polynomial $$20w^2 + 20w + 28w^3 + 31$$ by the binomial $$4w + 4$$ and express the result as quotient plus remainder over divisor.
2. **Rewrite the dividend in standard form:** Arrange terms in descending powers of $$w$$:
$$28w^3 + 20w^2 + 20w + 31$$
3. **Set up polynomial long division:** Divide $$28w^3 + 20w^2 + 20w + 31$$ by $$4w + 4$$.
4. **Divide leading terms:** $$\frac{28w^3}{4w} = 7w^2$$.
5. **Multiply divisor by $$7w^2$$:**
$$7w^2 \times (4w + 4) = 28w^3 + 28w^2$$
6. **Subtract:**
$$\left(28w^3 + 20w^2 + 20w + 31\right) - \left(28w^3 + 28w^2\right) = 0w^3 + (20w^2 - 28w^2) + 20w + 31 = -8w^2 + 20w + 31$$
7. **Divide leading terms:** $$\frac{-8w^2}{4w} = -2w$$.
8. **Multiply divisor by $$-2w$$:**
$$-2w \times (4w + 4) = -8w^2 - 8w$$
9. **Subtract:**
$$(-8w^2 + 20w + 31) - (-8w^2 - 8w) = 0w^2 + (20w + 8w) + 31 = 28w + 31$$
10. **Divide leading terms:** $$\frac{28w}{4w} = 7$$.
11. **Multiply divisor by $$7$$:**
$$7 \times (4w + 4) = 28w + 28$$
12. **Subtract:**
$$ (28w + 31) - (28w + 28) = 0w + 3 = 3$$
13. **Write the quotient and remainder:** Quotient is $$7w^2 - 2w + 7$$ and remainder is $$3$$.
14. **Express final answer:**
$$\frac{20w^2 + 20w + 28w^3 + 31}{4w + 4} = 7w^2 - 2w + 7 + \frac{3}{4w + 4}$$
15. **Check:** Multiply quotient by divisor and add remainder:
$$ (7w^2 - 2w + 7)(4w + 4) + 3 = 28w^3 + 20w^2 + 20w + 31$$ which matches the original dividend.
**Final answer:**
$$7w^2 - 2w + 7 + \frac{3}{4w + 4}$$