Use synthetic division to divide
2x^5 + 3x^4 + 25x^² − 1 by x + 3.
Also, why call it synthetic?
Why is it called synthetic division?
Umm, maybe because it is not "real" division, it is fake, or "synthetic", division.
[In building materials, "synthetic" signifies "not natural".]
[Things like this, definitions, I mean, you can search Google or Wikipedia or Yahoo.]
It is difficult to show here the form or layout of the synthetic division, but let me try.
"divide by (x+3)" means (x+3) may be a factor, and so (-3) may be a root, of the expression (2x^5 +3x^4 +25x^2 -1).
So, (-3) is the first divisor of sort.
-3...|2.....3.....0.....25.....0.....-1
............-6.....9....-27.....6....-18
________________________________
........2..-3.....9.....-2......6....-19
It is:
a) Bring down 2
b) -3*2 = -6
c) 3 +(-6) = -3
d) -3 *(-3) = 9
e) 0 +9 = 9
f) -3 *9 = -27
g) 25 +(-27) = -2
h) -3 *(-2) = 6
i) 0 +6 = 6
j) -3 *6 = -18
k) -1 +(-18) = -19
So the quotient is 2x^4 -3x^3 +9x^2 -2x +6, remainder (-19).
If you studied synthetic division, you may understand the above.
Not good to explain it here.
When I was at school I beleive that we were taught that synthetic division
was polynimial long division. However this is not the case synthetic division
is a short cut which is applicable when the divisor is a linear term.
This site provides more detail.
RonL