1. **Problem (b):** Simplify the expression $$(2w^3 - 4 - 8w - 3w^2 + w^6) + (w^2 - w - 2)$$ 2. **Step 1:** Write the expression grouping like terms: $$w^6 + 2w^3 - 3w^2 + w^2 - 8w - w - 4 - 2$$ 3. **Step 2:** Combine like terms: $$w^6 + 2w^3 - (3w^2 - w^2) - (8w + w) - (4 + 2) = w^6 + 2w^3 - 2w^2 - 9w - 6$$ --- 1. **Problem (c):** Divide $$\frac{t^4 - 17t^2 - 36t - 20}{t^2 - 3t - 10}$$ 2. **Step 1:** Factor the denominator: $$t^2 - 3t - 10 = (t - 5)(t + 2)$$ 3. **Step 2:** Perform polynomial long division: Divide $t^4$ by $t^2$ to get $t^2$. Multiply divisor by $t^2$: $$t^4 - 3t^3 - 10t^2$$ Subtract: $$\cancel{t^4} - 17t^2 - 36t - 20 - (\cancel{t^4} - 3t^3 - 10t^2) = 3t^3 - 7t^2 - 36t - 20$$ 4. Divide $3t^3$ by $t^2$ to get $3t$. Multiply divisor by $3t$: $$3t^3 - 9t^2 - 30t$$ Subtract: $$\cancel{3t^3} - 7t^2 - 36t - 20 - (\cancel{3t^3} - 9t^2 - 30t) = 2t^2 - 6t - 20$$ 5. Divide $2t^2$ by $t^2$ to get $2$. Multiply divisor by $2$: $$2t^2 - 6t - 20$$ Subtract: $$\cancel{2t^2} - 6t - 20 - (\cancel{2t^2} - 6t - 20) = 0$$ 6. **Answer:** $$t^2 + 3t + 2$$ --- 1. **Problem (d):** Divide $$\frac{x^3 + x^2y - xy^2 - y^3}{x - y}$$ 2. **Step 1:** Use polynomial long division or factor by grouping. 3. Factor numerator by grouping: $$x^3 + x^2y - xy^2 - y^3 = (x^3 + x^2y) - (xy^2 + y^3) = x^2(x + y) - y^2(x + y) = (x + y)(x^2 - y^2)$$ 4. Recognize difference of squares: $$x^2 - y^2 = (x - y)(x + y)$$ 5. So numerator: $$(x + y)(x - y)(x + y) = (x - y)(x + y)^2$$ 6. Divide numerator by denominator: $$\frac{(x - y)(x + y)^2}{x - y} = (x + y)^2$$ 7. **Answer:** $$(x + y)^2$$ --- 1. **Problem (e):** Divide $$\frac{x^4 - 2x^3y + 2x^2y^2 - 2xy^3 + y^4}{x^2 + y^2}$$ 2. **Step 1:** Try polynomial long division or factor numerator. 3. Notice numerator resembles expansion of $(x^2 - y^2)^2$ but signs differ. 4. Perform polynomial long division: Divide $x^4$ by $x^2$ to get $x^2$. Multiply divisor by $x^2$: $$x^4 + x^2y^2$$ Subtract: $$\cancel{x^4} - 2x^3y + 2x^2y^2 - 2xy^3 + y^4 - (\cancel{x^4} + x^2y^2) = -2x^3y + x^2y^2 - 2xy^3 + y^4$$ 5. Divide $-2x^3y$ by $x^2$ to get $-2xy$. Multiply divisor by $-2xy$: $$-2x^3y - 2xy^3$$ Subtract: $$\cancel{-2x^3y} + x^2y^2 - 2xy^3 + y^4 - (\cancel{-2x^3y} - 2xy^3) = x^2y^2 + y^4$$ 6. Divide $x^2y^2$ by $x^2$ to get $y^2$. Multiply divisor by $y^2$: $$x^2y^2 + y^4$$ Subtract: $$\cancel{x^2y^2} + y^4 - (\cancel{x^2y^2} + y^4) = 0$$ 7. **Answer:** $$x^2 - 2xy + y^2$$ --- 1. **Problem (f):** Divide $$\frac{x^3 - 4x^2y + 5xy^2 - 2y^3}{x - 2y}$$ 2. **Step 1:** Perform polynomial long division. 3. Divide $x^3$ by $x$ to get $x^2$. Multiply divisor by $x^2$: $$x^3 - 2x^2y$$ Subtract: $$\cancel{x^3} - 4x^2y + 5xy^2 - 2y^3 - (\cancel{x^3} - 2x^2y) = -2x^2y + 5xy^2 - 2y^3$$ 4. Divide $-2x^2y$ by $x$ to get $-2xy$. Multiply divisor by $-2xy$: $$-2x^2y + 4xy^2$$ Subtract: $$\cancel{-2x^2y} + 5xy^2 - 2y^3 - (\cancel{-2x^2y} + 4xy^2) = xy^2 - 2y^3$$ 5. Divide $xy^2$ by $x$ to get $y^2$. Multiply divisor by $y^2$: $$xy^2 - 2y^3$$ Subtract: $$\cancel{xy^2} - 2y^3 - (\cancel{xy^2} - 2y^3) = 0$$ 6. **Answer:** $$x^2 - 2xy + y^2$$