Determine whether the given binomial is a factor of x^3+13x^2+31x-45
x+5
is the binomial x+5 a factor of x^3+13x^2+31x-45
yes
no
Yes, x+5 is a factor of x^3+13x^2+31x-45.
To determine whether the binomial x+5 is a factor of x^3+13x^2+31x-45, we can use the Synthetic Division method.
1. Start by writing the coefficients of the polynomial in descending order:
x^3 + 13x^2 + 31x - 45
2. Now, bring down the coefficient of the highest power of x, which is 1:
1
3. Multiply the binomial, x+5, by the divisor 1 (the coefficient of x^3):
x+5
4. Add the results of the multiplications:
-(5)
5. Write the result below the next coefficient (13):
1
-(5)
6. Repeat the process by multiplying the new polynomial, x^2+(13-5)x-5, by the same divisor (x+5):
x+5
7. Add the results:
-5
10
8. Write the result below the next coefficient (31):
1
-(5)
10
9. Repeat the process one last time by multiplying the new polynomial, x^2+(13-5)x-5+(10), by (x+5):
x+5
10. Add the results:
-5
10
-10
11. Write the final result below the last coefficient (-45):
1
-(5)
10
-(10)
12. Examine the last number in the final row (-10). If it is zero, then the binomial x+5 is a factor of the polynomial. In this case, -10 is not zero, so x+5 is not a factor of x^3+13x^2+31x-45.
Therefore, the binomial x+5 is not a factor of x^3+13x^2+31x-45.