Add the following expressions using both the horizontal method and column method:
$4 x^{3} - 3 x^{2} - x + 3, 7 + 2 x + 5 x^{2} - 3 x^{3}, - 4 x^{2} - 9 x$ and $- 7 - x^{2} - x^{3}$

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Solution

Step 1: Adding them by the horizontal method
Horizontal method: Taking care of signs, write all expressions in a horizontal line and group like terms together. Now, add these like terms and get the required sum.
We have $4 x^{3} - 3 x^{2} - x + 3, 7 + 2 x + 5 x^{2} - 3 x^{3}, - 4 x^{2} - 9 x$ and $- 7 - x^{2} - x^{3}$
Now,
$(4 x^{3} - 3 x^{2} - x + 3) + (7 + 2 x + 5 x^{2} - 3 x^{3}) + (- 4 x^{2} - 9 x) + (- 7 - x^{2} - x^{3}) = 4 x^{3} - 3 x^{2} - x + 3 + 7 + 2 x + 5 x^{2} - 3 x^{3} - 4 x^{2} - 9 x - 7 - x^{2} - x^{3} = 4 x^{3} - 3 x^{3} - x^{3} - 3 x^{2} + 5 x^{2} - 4 x^{2} - x^{2} - x + 2 x - 9 x + 3 + 7 - 7 = (4 - 3 - 1) x^{3} + (- 3 + 5 - 4 - 1) x^{2} + (- 1 + 2 - 9) x + (3 + 7 - 7) = 0 x^{3} + (- 3) x^{2} + (- 8) x + 3 = - 3 x^{2} - 8 x + 3$
Step 2: Adding them by the column method
Column method: Write each expression in separate lines such that their like terms lie in the same vertical column. Taking care of signs, add each column.
$4 x^{3} - 3 x^{2} - x + 3 - 3 x^{3} + 5 x^{2} + 2 x + 7 0 - 4 x^{2} - 9 x + 0 + - x^{3} - x^{2} + 0 - 7- 3 x^{2} - 8 x + 3_$
Hence, the required sum is $- 3 x^{2} - 8 x + 3$

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