c) Subtract the sum of x 8x+2 and 4x? 3x-3 from 9x7x+10
Answers
x3 + 3x2 + 5x – 4) – (3x3 – 8x2 – 5x + 6)
(x3 + 3x2 + 5x – 4) – 1(3x3 – 8x2 – 5x + 6)
(x3 + 3x2 + 5x – 4) – 1(3x3) – 1 (–8x2) – 1(–5x) – 1(6)
x3 + 3x2 + 5x – 4 – 3x3 + 8x2 + 5x – 6
x3 – 3x3 + 3x2 + 8x2 + 5x + 5x – 4 – 6
–2x3 + 11x2 + 10x –10
And here's what the subtraction looks like, when going vertically:
\small{ \begin{array}{rrrr}x^3&+3x^2&+5x&-4\\-(3x^3&-8x^2&-5x&+6)\\ \hline\end{array} }
x
3
−(3x
3
+3x
2
−8x
2
+5x
−5x
−4
+6)
In the horizontal addition (above), you may have noticed that running the negative through the parentheses changed the sign on each and every term inside those parentheses. The shortcut when working vertically is to not bother writing in the subtaction sign or the parentheses; instead, write the second polynomial in the second row, and then just flip all the signs in that row, "plus" to "minus" and "minus" to "plus".
I'll change all the signs in the second row (shown in red below), and add down:
\small{ \begin{array}{rrrr}x^3&+3x^2&+5x&-4\\ \textcolor{red}{\textbf{--}}\,3x^3&\textcolor{red}{\textbf{+}}\,8x^2&\textcolor{red}{\textbf{+}}\,5x&\textcolor{red}{\textbf{--}}\,6\\ \hline -2x^3&+11x^2&+10x&-10\end{array} }
x
3
–3x
3
−2x
3
+3x
2
+8x
2
+11x
2
+5x
+5x
+10x
−4
–6
−10
Either way, I get the answer:
–2x3 + 11x2 + 10x – 10