x^3+3x^2-10x-24/ (x+4)
The remainder for this is zero. Correct?
http://www.purplemath.com/modules/remaindr.htm
You need to do some synthetic division.
I did and I arrived at 0. Were my calculations correct?
Correct!
You can also use the fact that
f(x)/(x-k) has a remainder of f(k).
In this case,
f(x)=x^3+3x^2-10x-24
k=-4
so evaluate
Remainder of f(x)/(x-k)
=Remainder of (x^3+3x^2-10x-24)/(x+4)
=f(-4)
=(-4)^3+3(-4)^2-10(-4)-24
=-64+48+40-24
=0
Thanks!
You're welcome!
To determine if the remainder is zero when dividing a polynomial by another polynomial, you can use synthetic division or long division.
Let's use synthetic division to divide the polynomial x^3 + 3x^2 - 10x - 24 by x + 4.
1. Set up the synthetic division like this:
-4 | 1 3 -10 -24
2. Bring down the coefficient of the highest power term, which is 1:
-4 | 1 3 -10 -24
_____________________
1
3. Multiply the divisor (-4) by the number at the bottom and write the result under the next coefficient:
-4 | 1 3 -10 -24
-4
_____________________
1 -4
4. Add the vertical column of numbers to get new coefficients:
-4 | 1 3 -10 -24
-4
_____________________
1 -4 -14
The final result of the synthetic division is a new polynomial with coefficients 1, -4, -14. This represents the quotient polynomial.
In this case, since the last number in the synthetic division is not zero, the remainder is not zero.
Therefore, the statement "The remainder for this is zero" is incorrect. The remainder is -14.