Most valuable coefficient - ll

f(x) =2(x-1)(x-3)(x-5)(x-7)+g(x)

g(x)=(x-7)*h(x) as (x-7) is a factor of f(x)

g(1)=(1-7)*(1 $\times$ 1)

g(3)=(3-7)*(3 $\times$ 3)

$\therefore h(x)=x^2$

$f(x) =2(x-1)(x-3)(x-5)(x-7) +(x-7)x^2$

$f(8)=2\times7\times5\times3\times1 +64 =274.$

$f(x) = 2(x-1)(x-3)(x-5)(x-7) + g(x)$ where g(x) is a unique polynomial of degree 3 satisfying the above conditions.
Consider,
$p_{1}(x) = (x-3)(x-5)(x-7)$
$p_{1}(1) = -48$
$p_{3}(x) = (x-1)(x-5)(x-7)$
$p_{3}(3) = 16$
$p_{5}(x) = (x-1)(x-3)(x-7)$
$p_{5}(5) = -16$
$p_{7}(x) = (x-1)(x-3)(x-5)$
$p_{7}(7) = 48$
Then let, $h(x) = \dfrac{f(1) \times p_{1}(x)}{p_{1}(1)} + \dfrac{f(3) \times p_{3}(x)}{p_{3}(3)} + \dfrac{f(5) \times p_{5}(x)}{p_{5}(5)} + \dfrac{f(7) \times p_{7}(x)}{p_{7}(7)}$
$h(x)$ is a polynomial satisfying all the required conditions. Since the polynomial is unique, $h(x) = g(x)$

$\therefore g(x) = \dfrac{f(1) \times p_{1}(x)}{p_{1}(1)} + \dfrac{f(3) \times p_{3}(x)}{p_{3}(3)} + \dfrac{f(5) \times p_{5}(x)}{p_{5}(5)} + \dfrac{f(7) \times p_{7}(x)}{p_{7}(7)}$

$f(x) = 2(x-1)(x-3)(x-5)(x-7) + \dfrac{f(1) \times p_{1}(x)}{p_{1}(1)} + \dfrac{f(3) \times p_{3}(x)}{p_{3}(3)} + \dfrac{f(5) \times p_{5}(x)}{p_{5}(5)} + \dfrac{f(7) \times p_{7}(x)}{p_{7}(7)}$
$f(x) = 2(x-1)(x-3)(x-5)(x-7) + \dfrac{-6 \times (x-3)(x-5)(x-7) }{-48} + \dfrac{-36 \times (x-1)(x-5)(x-7)}{16} + \dfrac{-50 \times (x-1)(x-3)(x-7))}{-16} + \dfrac{0 \times (x-1)(x-3)(x-5)}{48}$

$f(8) = 2 \times 105 + \dfrac{6 \times 15 }{48} - \dfrac{36 \times 21}{16} + \dfrac{50 \times 35}{16} + 0$

$f(8) = 210 + \dfrac{15}{8} + \dfrac{994}{16}$

$f(8) = 210 + \dfrac{30+994}{16}$
$f(8) = 210 + \dfrac{1024}{16} = 210 + \dfrac{2^{10}}{2^4} = 210 + 2^6 = 210 + 64 = 274$