Given that the quadratic polynomial f(x)=3x^2+ax+b has real coefficients and f(x)=0 has a complex root 8+5i, what is the value of a+b?
The other root must be 8-5i, so
f(x) = 3(x-(8+5i))(x-(8-5i))
= 3((x-8)^2 + 5^2)
= 3(x^2-16x+89)
= 3x^2 - 48x + 267
a+b = 239
219 sir
To find the value of a+b, we need to use the fact that the polynomial f(x)=3x^2+ax+b has real coefficients and that f(x)=0 has a complex root 8+5i.
We know that complex roots of a polynomial with real coefficients occur in conjugate pairs. So, if 8+5i is a root, then its conjugate, 8-5i, must also be a root.
To find the sum of the roots, we can use Vieta's formulas for a quadratic polynomial. According to Vieta's formulas, the sum of the roots of a quadratic polynomial ax^2+bx+c=0 is given by -b/a.
In this case, the sum of the roots 8+5i and 8-5i will be (8+5i) + (8-5i) = 16. Since the sum of the roots is given by -b/a, we have -b/a = 16.
We also know that the coefficient of x in the quadratic polynomial is a. Since the sum of the roots is given by -b/a, it means that a = -b/16.
Since a and b are both real coefficients, we can conclude that a = -b/16 is a real equation.
Now, we can substitute the value of a in terms of b into the original equation and solve for b. From f(x) = 3x^2 + ax + b, we substitute a = -b/16 to get: f(x) = 3x^2 - (b/16)x + b.
We know that the quadratic equation f(x) = 3x^2 - (b/16)x + b = 0 has the complex root 8+5i. Therefore, this equation satisfies the quadratic formula x = (-b/32) ± √((b^2/256) - 3b).
Since the quadratic formula satisfies the complex root, the discriminant of the quadratic equation, (b^2/256) - 3b, must be negative.
Simplifying the discriminant inequality, we get b^2 - 768b < 0.
Now, we can solve this inequality to find the range of b values. Factoring out b, we get b(b - 768) < 0. The zeros of the inequality occur at b = 0 and b = 768.
Since we are looking for real coefficients, we know that b cannot equal 0, so we have 0 < b < 768.
Next, we substitute b into the equation a = -b/16 to find the corresponding values of a. For any value of b within the range 0 < b < 768, we can calculate the corresponding value of a using the equation a = -b/16.
Finally, once we have the values of a and b, we can find the sum a+b to get the answer.