Solutions to add and subtract polynomials are presented along with detailed explnations.

## Solutions to the Questions in Add and Subtract Polynomials

1. Add and Subtract the like terms.

1. ) $2x - 2x + 9x = (2 - 2 +9) x = 9 x$

2. ) $-x^2 + 3x^2 + x^2 = (-1 + 3 + 1)x^2 = 3x^2$  

3. ) $-x y + \dfrac{2}{3} x y + \dfrac{1}{2}x y = (-1 + \dfrac{2}{3} + \dfrac{1}{2}) xy = (-\dfrac{6}{6} + \dfrac{4}{6} + \dfrac{3}{6}) xy = \dfrac{1}{6} x y$

4. ) $0.2 x^3 + 2 x^3 - 0.5 x^3 = (0.2 + 2 - 0.5)x^3 = 1.7 x^3 = \dfrac{17}{10} x^3$

5. ) $x -0.3 x - \dfrac{1}{5}x = (1 -0.3 - \dfrac{1}{5})x = (\dfrac{10}{10} - \dfrac{3}{10} - \dfrac{2}{10} ) x = \dfrac{1}{2} x = 0.5 x$

2. Add and Subtract the following polynomials.

1. )
$\begin{split} (2x^2 - 2x + 1) + (x + 5) &= \color{red}{+1}(2x^2 - 2x + 1) + \color{red}{+1}(x + 5) \\\\ &= 2x^2 - 2x + 1 + x + 5 \\\\ &= 2x^2 +(-2x+x) + (1+ 5) \\\\ &= 2x^2 - x + 6 \end{split}$

2. )
$\begin{split} (- 4x^3 - 2x + 1) - ( - x^3 - 5 x) &= \color{red}{+1}(- 4x^3 - 2x + 1) + \color{red}{-1}(- x^3 - 5 x) \\\\ &= - 4x^3 - 2x + 1 + x^3 + 5 x \\\\ &= (- 4x^3+x^3) + (-2x+5x) + 1 \\\\ &= -3x^3+3x + 1 \end{split}$

3. )
$\begin{split} - (2x^3 - 2x^2 + 1) + ( - x^3 - 5 x^2) &= \color{red}{-1} (2x^3 - 2x^2 + 1) \color{red}{+1} ( - x^3 - 5 x^2) \\\\ &= -2x^3 + 2x^2 - 1 - x^3 - 5 x^2 \\\\ &= (-2x^3 - x^3) + ( 2x^2-5x^2) + 1 \\\\ &= -3x^3-3x^2-1 \end{split}$

4. )
$\begin{split} - ( - x^4 y - 2 x^2 - 9 ) - ( - y x^4 - 5 x^2 + 1) &= \color{red}{-1} ( - x^4 y - 2 x^2 - 9 ) \color{red}{-1} ( - y x^4 - 5 x^2 + 1) \\\\ &= x^4 y + 2 x^2 + 9 + y x^4 + 5 x^2 - 1 \\\\ &= (x^4 y + y x^4) + (2 x^2 + 5 x^2) + (9-1) \\\\ &= 2 x^4 y + 7x^2 + 8 \end{split}$

5. )
$\begin{split} ( - x^2 - 2 x ) - ( - x^2 - 5 x + 3) + ( x^2 - 4 ) &= \color{red}{+1}( - x^2 - 2 x ) \color{red}{-1} ( - x^2 - 5 x + 3) \color{red}{+1} ( x^2 - 4 ) \\\\ &= - x^2 - 2 x + x^2 + 5 x - 3 + x^2 - 4 \\\\ &= (- x^2 + x^2 + x^2)+(-2x+5x) +(-3-4) \\\\ &= x^2 + 3x-7 \end{split}$

6. )
$\begin{split} ( x^3 - 2x^2 + 3) - ( \dfrac{1}{4}x^3 + \dfrac{1}{2} x^2 - \dfrac{1}{3}) &= \color{red}{+1}( x^3 - 2x^2 + 3) \color{red}{-1} ( \dfrac{1}{4}x^3 + \dfrac{1}{2} x^2 - \dfrac{1}{3}) \\\\ &= x^3 - 2x^2 + 3 -\dfrac{1}{4}x^3 - \dfrac{1}{2} x^2 + \dfrac{1}{3} \\\\ &= ( 1 -\dfrac{1}{4}) x^3 +(-2- \dfrac{1}{2})x^2 +(3+ \dfrac{1}{3}) \\\\ &= \dfrac{3}{4} x^3 -\dfrac{5}{2} x^2 + \dfrac{10}{3} \end{split}$