In (8b+2b^3)/(b-1), Do you have to put a zero coefficient in it?
yes, if you want to perform the algebraic long division, or synthetic division.
arrange it like
2b^3 + 0x^2 + 8b ÷ (b-1)
with either method you should end up with
2b^2 + 2b + 10 with a remainder of 10
Are you sure it isn't 10b?
I am sure:
2b^3 + 0x^2 + 8b ÷ (b-1)
= (2b^2 + 2b + 10) + 10/(b-1)
check :
add the terms in my answer,
(2b^2 + 2b + 10) + 10/(b-1)
= [(2b^2 + 2b + 10)(b-1) + 10]/(b-1)
= (2b^3 + 8b) ÷ (b-1) or the original question.
another way to check answers like this is to take any value of b, say b=2
and sub it in the original and in the answer.
If you get the same result, there is a very good probablility that the question was done correctly.
This does not prove that it is right, but if you don't get the same result, then you know for sure that your answer is wrong.
To determine if you need to include a zero coefficient in the expression (8b+2b^3)/(b-1), you can simplify the expression and check if there are any missing terms.
First, let's simplify the expression by factoring out the greatest common factor (GCF) from the numerator:
8b + 2b^3 = 2b(4 + b^2)
Now, let's factor the denominator (b-1) by using difference of squares:
b-1 = (b-1)(b+1)
Next, we can cancel out the common factors between the numerator and denominator:
(2b(4 + b^2))/((b-1)(b+1))
So, after simplifying the expression, we don't have any zero coefficients. However, it is important to note that zero coefficients can appear in certain situations depending on the original expression or equation. In this particular case, no zero coefficients are involved.