Perform the indicated operations and simplify the result:

(a) $(7x^{3}-6x^{2}+4x+9)+(x^{3}+4x^{2}-2)$

(b) $(-3x^{4}-8x)-(5x^{3}-3x+\sqrt{2}) $

Solution

(a) $\begin{aligned} (7x^{3}-6x^{2}+4x+9)+(x^{3}+4x^{2}-2) & =(7+1)x^{3}+(-6+4)x^{2} &\qquad +(4+0)x+(9-2) & =8x^{3}-2x^{2}+4x+7 \end{aligned}$ (b) $\begin{aligned} (-3x^{4}-8x)-(5x^{3}-3x+\sqrt{2}) & =(-3-0)x^{4}+(0-5)x^{3} & \qquad +(-8-(-3))x+(0-\sqrt{2}) & =-3x^{4}-5x^{3}-5x-\sqrt{2} \end{aligned}$