Three roots of f(x)=x4−2x3+ax2+bx+c are −5, −3 and 4. What is the value of a+b+c?
To find the value of a+b+c, we need to use the given roots of the polynomial equation f(x)=x^4−2x^3+ax^2+bx+c.
The given roots are -5, -3, and 4. We can use these roots to form linear factors of the polynomial equation.
For a root of -5, we have (x+5) as a factor.
For a root of -3, we have (x+3) as a factor.
For a root of 4, we have (x-4) as a factor.
By multiplying these factors together, we can obtain the polynomial equation:
f(x) = (x+5)(x+3)(x-4)(x-r), where 'r' is the fourth root.
Expanding this equation, we get:
f(x) = (x^2 + 8x + 15)(x^2 - 4x - 12)(x-r)
Comparing this expanded equation with the given polynomial equation f(x)=x^4−2x^3+ax^2+bx+c, we can equate the coefficients.
For the x^3 term, we have: -2 = -(4 + r) [coefficient of x^3 in both equations]
Solving this equation, we find: r = -6
So now we have the fourth root, which is -6.
Replacing 'r' in the expanded equation, we have:
f(x) = (x^2 + 8x + 15)(x^2 - 4x - 12)(x+6)
Now we can find the values of a, b, and c by equating the coefficients of the polynomial equation.
The coefficient of x^2 in f(x) is obtained by multiplying the corresponding terms in the three factors:
a = (1)(1)(1) + (-4)(1)(1) + 15(-4)
a = 1 - 4 - 60
a = -63
The coefficient of x is obtained by multiplying the corresponding terms in the three factors:
b = (1)(8)(1) + (-4)(8)(1) + 15(-6)
b = 8 - 32 - 90
b = -114
The constant term is obtained by multiplying the constant terms in the three factors:
c = (15)(-12)(1)(6)
c = -1080
Therefore, the value of a+b+c is:
(-63) + (-114) + (-1080) = -1257
Hence, the value of a+b+c in the given polynomial equation is -1257.