Page 60 - SM inner class 7.cdr

Adding Polynomials that, for simple additions, horizontal addition
Adding polynomials is just a matter of (so you don't have to rewrite the problem) is
combining like terms, with some order of probably simplest, but, once the polynomials
operations considerations. As long as you're get complicated, vertical is probably safest bet
careful with the minus signs, and don't confuse (so you don't "drop", or lose, terms and minus
addition and multiplication, you should do fine. signs).
There are a couple formats for adding and † Sim plify 3 2
2
3
4
+
subtracting, and they are just like, you have (3x + 3x – 4x + 5 ) (x – 2x + x – )
learnt in earlier times, when you were adding We can add hor i zon tally:
3
2
4
+
and subtracting just plain old numbers. First, (3x + 3x 2 – 4x + 5 ) (x 3 – 2x + x – )
you learned addition "horizontally", like this: = 3x 3 + 3x – 4x + + x – 2x 2 + x – 4
3
2
5
+
6 3 = 9. You can add polynomials in the same = 3x 3 + x 3 + 3x – 2x – 4x + x + 5 4
2
2
–
way, grouping like terms and then simplifying. 3 2
= 4x +1x – 3x +1
† Sim plify (2x + 5y ) (3x – 2y )
+
...or ver ti cally:
We'll clear the parentheses, group like terms, 3x + 3x - 4x + 5
2
3
and then simplify:
3
2
x - 2 x + x - 4
(2x + 5y ) (3x – 2y )
+
2
3
1
4x + 1x - 3x +
= 2x + 5y + 3x – 2y
Either way, We get the same answer:
= 2x + 3x + 5y – 2y (By grouping) 3 2
4x + 1x – 3x + 1.
= 5x + 3y
Note that each column in the vertical addition
Horizontal addition works fine for simple above contains only one degree of x: the first
examples. But when you were adding plain old 3
column was the x column, the second column
numbers, you didn't generally try to add 432 was the x 2 column, the third column was
and 246 horizontally; instead, you would the x column, and the fourth column was the
"stack" them vertically, one on top of the other, constants column. This is analogous to having
and then add down the columns
a thousands column, a hundreds column, a
4 3 2 tens column, and a ones column when doing
+ 2 4 6 strictly-numerical addition.
3
+
4
6 7 8 † Sim plify (7x 2 – x – ) (x 2 – 2x – )
You can do the same thing with polynomials. +(–2x 2 + 3x + )
5
This is how the above simplification exercise It's perfectly okay to have to add three or more
looks when it is done "vertically"
polynomials at once. We'll just go slowly and do
+
† Sim plify (2x + 5y ) (3x – 2y ) each step thoroughly, and it should work out
We'll put each variable in its own column; in right.
this case, the first column will be the x-column, Adding hor i zon tally:
and the second column will be the y-column: 2 2
+
4
(7x – x – ) (x – 2x – )
3
2x + 5y +(–2x 2 + 3x + )
5
3x - 2y
2
2
–
= 7x – x – 4 + x – 2x – 3 + 2x 2 + 3x + 5
5x + 3y
2
2
= 7x 2 +1x – 2x – 1x – 2x + 3x – 4 – 3 + 5
We get the same solution vertically as we got = 8x – 2x – 3x + 3x – 7 + 5
2
2
horizontally: 5x + 3y.
2
= 6x – 2
The format you use, horizontal or vertical, is a
matter of taste (unless the instructions Note the 1's in the third line. Any time you
explicitly tell you otherwise). Given a choice, have a variable without a coefficient, there is
an "understood" 1 as the coefficient. If you find
you should use whichever format that you're
it helpful to write that 1 in, then do so.
more comfortable and successful with. Note
60
Mathematics In Focus - 7