This is a symplify - the order I have is wrong?

Question

(4t-5w+3)(5t-2w+7)

simplified:
20t^2-33wt+43t+10w^2-41w+21

Is my order correct?

Answer 1

Original equation:
(4t - 5w + 3) (5t - 2w + 7)

Equivalent equation:
4t (5t - 2w + 7) - 5w (5t - 2w + 7) + 3 (5t - 2w + 7)

Now multiply through:
20t² - 8tw + 28t - 25tw + 10w² - 35w + 15t - 6w + 21

Order the terms to put like terms together:
20t² + (-8tw -25tw) + (28t + 15t) + 10w² + (-35w -6w) + 21


You can also do this with matrices:

... t w 1
t
w
1

... t w 1
t t², tw, t
w tw, w², w
1 t, w, 1

... 5, -2, 7
4
-5
3

... 5, -2, 7
4 20, -8, 28
-5 -25, 10, -35
3 15, -6, 21

20t² + (-25tw -8tw) + (15t + 28t) + 10w² + (-6w -35w) + 21

Simplify:
20t² + (-33tw) + (43t) + 10w² + (-41w) + 21

Final answer:
20t² - 33tw + 43t +10w² - 41w + 21

Note, it doesn't matter if you say tw, or wt, or if you put t² before w², etc. as long as you have all the terms.

Either way I'd say you got the right answer!

Answer 2

Your answer is perfectly correct. There is no nice way to order the terms so any order should still work.

Hope this helps. Good Luck.

Answer 3

the order of terms in a polynomial is insignificant