1. **Problem statement:** Find $(f+g)(x)$ where $$f(x) = 6x^4 - x^3 + 3x^2 - 4x + 7$$ $$g(x) = -4x^4 + x^3 - 2x^2 + 5x - 5$$ 2. **Formula:** The sum of two functions is given by $$(f+g)(x) = f(x) + g(x)$$ 3. **Add corresponding terms:** $$6x^4 + (-4x^4) = 6x^4 - 4x^4 = 2x^4$$ $$-x^3 + x^3 = 0$$ $$3x^2 + (-2x^2) = 3x^2 - 2x^2 = x^2$$ $$-4x + 5x = x$$ $$7 + (-5) = 2$$ 4. **Combine all:** $$(f+g)(x) = 2x^4 + 0 + x^2 + x + 2 = 2x^4 + x^2 + x + 2$$ 5. **Evaluate $(f+g)(10)$:** $$2(10)^4 + (10)^2 + 10 + 2 = 2(10000) + 100 + 10 + 2 = 20000 + 100 + 10 + 2 = 20112$$ --- 6. **Problem statement:** Find $(v-w)(y)$ where $$v(y) = -3y^4 - 5y^3 - 12y^2 + 8y - 10$$ $$w(y) = -7y^4 - 4y^3 + 10y^2 - 2y + 6$$ 7. **Formula:** The difference of two functions is $$(v-w)(y) = v(y) - w(y)$$ 8. **Subtract corresponding terms:** $$-3y^4 - (-7y^4) = -3y^4 + 7y^4 = 4y^4$$ $$-5y^3 - (-4y^3) = -5y^3 + 4y^3 = -y^3$$ $$-12y^2 - 10y^2 = -12y^2 - 10y^2 = -22y^2$$ $$8y - (-2y) = 8y + 2y = 10y$$ $$-10 - 6 = -16$$ 9. **Combine all:** $$(v-w)(y) = 4y^4 - y^3 - 22y^2 + 10y - 16$$ 10. **Evaluate $(w-v)(-6)$:** First find $(w-v)(y) = w(y) - v(y)$: $$w(y) - v(y) = (-7y^4 - 4y^3 + 10y^2 - 2y + 6) - (-3y^4 - 5y^3 - 12y^2 + 8y - 10)$$ Subtract term by term: $$-7y^4 - (-3y^4) = -7y^4 + 3y^4 = -4y^4$$ $$-4y^3 - (-5y^3) = -4y^3 + 5y^3 = y^3$$ $$10y^2 - (-12y^2) = 10y^2 + 12y^2 = 22y^2$$ $$-2y - 8y = -10y$$ $$6 - (-10) = 6 + 10 = 16$$ So $$(w-v)(y) = -4y^4 + y^3 + 22y^2 - 10y + 16$$ Evaluate at $y = -6$: $$-4(-6)^4 + (-6)^3 + 22(-6)^2 - 10(-6) + 16$$ Calculate powers: $$(-6)^4 = 1296$$ $$(-6)^3 = -216$$ $$(-6)^2 = 36$$ Substitute: $$-4(1296) + (-216) + 22(36) + 60 + 16 = -5184 - 216 + 792 + 60 + 16$$ Sum stepwise: $$-5184 - 216 = -5400$$ $$-5400 + 792 = -4608$$ $$-4608 + 60 = -4548$$ $$-4548 + 16 = -4532$$ **Final answers:** $$(f+g)(x) = 2x^4 + x^2 + x + 2$$ $$(f+g)(10) = 20112$$ $$(v-w)(y) = 4y^4 - y^3 - 22y^2 + 10y - 16$$ $$(w-v)(-6) = -4532$$