To add or subtract polynomials, we add or subtract the coefficients of like terms.

Example 1.8 . 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) \[\begin{align} (7x^{3}-6x^{2}+4x+9)+(x^{3}+4x^{2}-2) & =(7+1)x^{3}+(-6+4)x^{2}\\ &\qquad +(4+0)x+(9-2)\\ & =8x^{3}-2x^{2}+4x+7 \end{align}\] 

 

(b) \[\begin{align} (-3x^{4}-8x)-(5x^{3}-3x+\sqrt{2}) & =(-3-0)x^{4}+(0-5)x^{3}\\ & \qquad +(-8-(-3))x+(0-\sqrt{2})\\ & =-3x^{4}-5x^{3}-5x-\sqrt{2} \end{align}\]