Solve f(x)=x^50/50+x^49/49+x^48/48+--+x^2/2+x+1

$$f ( x ) = \frac { x ^ { 50 } } { 50 } + \frac { x ^ { 49 } } { 49 } + \frac { x ^ { 48 } } { 48 } + - - + \frac { x ^ { 2 } } { 2 } + x + 1$$

Evaluate

$$\frac{x^{50}}{50}+\frac{x^{49}}{49}+\frac{x^{48}}{48}+\frac{x^{2}}{2}+x+1$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 50 and 49 is 2450. Multiply \frac{x^{50}}{50} times \frac{49}{49}. Multiply \frac{x^{49}}{49} times \frac{50}{50}.

$$\frac{49x^{50}}{2450}+\frac{50x^{49}}{2450}+\frac{x^{48}}{48}-\left(-\frac{x^{2}}{2}\right)+x+1$$

Since \frac{49x^{50}}{2450} and \frac{50x^{49}}{2450} have the same denominator, add them by adding their numerators.

$$\frac{49x^{50}+50x^{49}}{2450}+\frac{x^{48}}{48}-\left(-\frac{x^{2}}{2}\right)+x+1$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2450 and 48 is 58800. Multiply \frac{49x^{50}+50x^{49}}{2450} times \frac{24}{24}. Multiply \frac{x^{48}}{48} times \frac{1225}{1225}.

$$\frac{24\left(49x^{50}+50x^{49}\right)}{58800}+\frac{1225x^{48}}{58800}-\left(-\frac{x^{2}}{2}\right)+x+1$$

Since \frac{24\left(49x^{50}+50x^{49}\right)}{58800} and \frac{1225x^{48}}{58800} have the same denominator, add them by adding their numerators.

$$\frac{24\left(49x^{50}+50x^{49}\right)+1225x^{48}}{58800}-\left(-\frac{x^{2}}{2}\right)+x+1$$

Do the multiplications in 24\left(49x^{50}+50x^{49}\right)+1225x^{48}.

$$\frac{1176x^{50}+1200x^{49}+1225x^{48}}{58800}-\left(-\frac{x^{2}}{2}\right)+x+1$$

To add or subtract expressions, expand them to make their denominators the same. Multiply x+1 times \frac{58800}{58800}.

$$\frac{1176x^{50}+1200x^{49}+1225x^{48}}{58800}-\left(-\frac{x^{2}}{2}\right)+\frac{58800\left(x+1\right)}{58800}$$

Since \frac{1176x^{50}+1200x^{49}+1225x^{48}}{58800} and \frac{58800\left(x+1\right)}{58800} have the same denominator, add them by adding their numerators.

$$\frac{1176x^{50}+1200x^{49}+1225x^{48}+58800\left(x+1\right)}{58800}-\left(-\frac{x^{2}}{2}\right)$$

Do the multiplications in 1176x^{50}+1200x^{49}+1225x^{48}+58800\left(x+1\right).

$$\frac{1176x^{50}+1200x^{49}+1225x^{48}+58800x+58800}{58800}-\left(-\frac{x^{2}}{2}\right)$$

Multiply -1 and -1 to get 1.

$$\frac{1176x^{50}+1200x^{49}+1225x^{48}+58800x+58800}{58800}+\frac{x^{2}}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 58800 and 2 is 58800. Multiply \frac{x^{2}}{2} times \frac{29400}{29400}.

$$\frac{1176x^{50}+1200x^{49}+1225x^{48}+58800x+58800}{58800}+\frac{29400x^{2}}{58800}$$

Since \frac{1176x^{50}+1200x^{49}+1225x^{48}+58800x+58800}{58800} and \frac{29400x^{2}}{58800} have the same denominator, add them by adding their numerators.

$$\frac{1176x^{50}+1200x^{49}+1225x^{48}+58800x+58800+29400x^{2}}{58800}$$

Factor

$$\frac{1176x^{50}+1200x^{49}+1225x^{48}+29400x^{2}+58800x+58800}{58800}$$

Solution Steps

Factor out \frac{1}{58800}. Polynomial 1176x^{50}+1200x^{49}+1225x^{48}+29400x^{2}+58800x+58800 is not factored since it does not have any rational roots.

$$\frac{1176x^{50}+1200x^{49}+1225x^{48}+29400x^{2}+58800x+58800}{58800}$$