Determine whether the given binomial is a factor of x^3+13x^2+31x-45
x+5
To check if x+5 is a factor of x^3+13x^2+31x-45, we can use synthetic division.
Using synthetic division:
-5 | 1 13 31 -45
|______ -5 10 -55
1 8 41 -100
The result of the synthetic division is 1x^2 + 8x + 41 - 100/(x+5).
Since the remainder is not equal to zero, x+5 is not a factor of x^3+13x^2+31x-45.
To determine whether the binomial x+5 is a factor of the polynomial x^3+13x^2+31x-45, we can use the Factor Theorem.
Step 1: Set up the synthetic division.
-5 | 1 13 31 -45
--------------------------------
1 -8 11 0
Step 2: Interpret the results.
The numbers in the bottom row represent the coefficients of the resulting polynomial after dividing. The last number in the bottom row is zero, which means the binomial x+5 is a factor of the polynomial x^3+13x^2+31x-45.
Therefore, x+5 is a factor of x^3+13x^2+31x-45.