solve the problem 2xcubed-17xsquared+31x+20 given that 5 is a zero of f(x)= 2xcubed-17xsquared+31x+20
you can use lobng division or synthetic division
2;-2
To solve the problem, we can use synthetic division.
First, let's write down the polynomial and the given zero:
f(x) = 2x^3 - 17x^2 + 31x + 20
Zero: 5
Start by writing the coefficients of the polynomial in order:
2 -17 31 20
Now, set up the synthetic division table:
5 | 2 -17 31 20
Bring down the first coefficient (which is 2):
5 | 2 -17 31 20
10
Multiply the zero (5) by the first coefficient (2) and write the result below the second coefficient (-17):
5 | 2 -17 31 20
10
Add the result to the second coefficient (-17) and write the sum below the third coefficient (31):
5 | 2 -17 31 20
10 -35
Repeat the process by multiplying the zero (5) by the new result (-35) and write it below the third coefficient (31):
5 | 2 -17 31 20
10 -35
75
Add the new result to the fourth coefficient (20) and write the sum below:
5 | 2 -17 31 20
10 -35
75 95
The final row of numbers represents the coefficients of the resulting quadratic equation, where the last number (95) is the constant term, the second-to-last number (75) is the coefficient of the linear term, and the third-to-last number (-35) is the coefficient of the squared term. The remaining two terms represent the quadratic equation:
2x^2 - 35x + 75
Therefore, the original polynomial can be factored as:
f(x) = (x - 5)(2x^2 - 35x + 75)
Now, we need to solve the quadratic equation (2x^2 - 35x + 75) to find the other two zeros. We can either factor it further or use the quadratic formula.