I don't understand missing terms in algebra

Question

I come across a brief introduction to synthetic division in the book (Algebra I for dummies) anyways

I found out that I have to insert zero in case of missing term so in this example

7x^3 + 6x - 8 / (x-4)

I have to insert 0x^2 to make the synthetic division work

my question is why?

why is it missing in the first place

do all expressions in algebra have to be exact decreasing power?

isn't there a valid expressions with differences in exponents like x^3-x^10?

please note am a beginner also self-taught

thank you

Answer 1

It's similar to arithmetic

multiply 102 by 3

102

3

=

306

1 is the digit for 100's which means 100

0 is the missing digit for 10's or tens it means 0(10) = 0

2 is the digit for ones, 2 means 2(1) = 2

to multiply by 3

you have to also keep the 0 for the tens' place

it's standard to write numbers in order, on left the highest number the 100's place, then the 10's then the 1's

same with algebra. the 100's place in arithmetic is comparable to x^2 in algebra where x= 10, x^2 =100

then then 10s place is comparable to x, and the constant term is for ones

multiplication is the inverse of division, and vice versa.

you could write 3x^2 -x^10, if you need to for a given problem, but it's not standard and may confuse people reading it, as they expect to see highest power first.

but that's comparaable to writing 300-10,000,000,000 in arithmetic

or worse 3000000000000001 when you really mean 10,000,000,300 in the standard way to write integers.

10,000,000,300 = 10^10 + 3(10^2) or in scientific notation 1.0 x 10^11 + 3.0 x 10^2

= 1.0 x 10^11 + 0.00000003 x 10^11

to be able to add them it's like converting it so you're adding apples to apples, not apples + oranges

= 1.000000003 x 10^11 you need those zeros as place markers to make it correct