Question

$f(x) = x^2 + ax + b$, has one of its zeroes as $\frac{4+3\sqrt{3}}{2+\sqrt{3}}$ where a and b are integers.

$g(x) = x^4 + 2x^3 - 10x^2 + 4x - 10$ is a biquadratic such that $g(\frac{4+3\sqrt{3}}{2+\sqrt{3}}) = c\sqrt{3} + d$, where c and d are integers.

let $M = |a| + |b| + |c| + |d|$.

Then what is the Sum of the AP with first term 'b', common difference 'd' and no. of terms being M.

Answer 1

-351

Answer 2

1. Simplify the given root: Let $x_0 = \frac{4+3\sqrt{3}}{2+\sqrt{3}}$. Rationalize the denominator: $x_0 = \frac{(4+3\sqrt{3})(2-\sqrt{3})}{(2+\sqrt{3})(2-\sqrt{3})} = \frac{8 - 4\sqrt{3} + 6\sqrt{3} - 9}{2^2 - (\sqrt{3})^2} = \frac{-1 + 2\sqrt{3}}{4-3} = -1 + 2\sqrt{3}$.

2. Find coefficients $a$ and $b$ for $f(x)$: $f(x) = x^2 + ax + b$, where $a, b \in \mathbb{Z}$. Since $-1 + 2\sqrt{3}$ is a root and coefficients are integers, its conjugate $-1 - 2\sqrt{3}$ must also be a root. Sum of roots: $(-1 + 2\sqrt{3}) + (-1 - 2\sqrt{3}) = -2$. From Vieta's formulas, sum of roots $= -a$. So, $-a = -2 \implies a = 2$. Product of roots: $(-1 + 2\sqrt{3})(-1 - 2\sqrt{3}) = (-1)^2 - (2\sqrt{3})^2 = 1 - 12 = -11$. From Vieta's formulas, product of roots $= b$. So, $b = -11$. Thus, $f(x) = x^2 + 2x - 11$.

3. Evaluate $g(x)$ at the root $x_0$: Let $x = x_0 = -1 + 2\sqrt{3}$. From $f(x)=0$, we have $x^2 + 2x - 11 = 0$, which implies $x^2 = -2x + 11$. We can divide $g(x)$ by $x^2 + 2x - 11$: $g(x) = x^4 + 2x^3 - 10x^2 + 4x - 10$. Using polynomial division or substitution: $g(x) = (x^2 + 1)(x^2 + 2x - 11) + (2x + 1)$. Since $x_0^2 + 2x_0 - 11 = 0$: $g(x_0) = (x_0^2 + 1)(0) + (2x_0 + 1) = 2x_0 + 1$. Substitute $x_0 = -1 + 2\sqrt{3}$: $g(x_0) = 2(-1 + 2\sqrt{3}) + 1 = -2 + 4\sqrt{3} + 1 = 4\sqrt{3} - 1$.

4. Find coefficients $c$ and $d$: We are given $g(x_0) = c\sqrt{3} + d$, where $c, d \in \mathbb{Z}$. Comparing $4\sqrt{3} - 1$ with $c\sqrt{3} + d$, we get $c = 4$ and $d = -1$.

5. Calculate $M$: $M = |a| + |b| + |c| + |d|$ $M = |2| + |-11| + |4| + |-1| = 2 + 11 + 4 + 1 = 18$.

6. Calculate the sum of the AP: The AP has: First term = $b = -11$. Common difference = $d = -1$. Number of terms = $M = 18$. The sum of an AP is $S_n = \frac{n}{2}[2 \times \text{first term} + (n-1) \times \text{common difference}]$. $S_{18} = \frac{18}{2}[2(-11) + (18-1)(-1)]$ $S_{18} = 9[-22 + (17)(-1)]$ $S_{18} = 9[-22 - 17]$ $S_{18} = 9[-39]$ $S_{18} = -351$.