Use the remainder theorem to evaluate the polynomial for the given values of $$ x $$. $$ f(x)=2 x^{4}-17 x^{3}+30 x^{2}+64 x+10 $$

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To evaluate the polynomial $$ f(x) = 2x^{4} - 17x^{3} + 30x^{2} + 64x + 10 $$ for the given values of $$ x $$ using the remainder theorem, we need to find the remainder when the polynomial is divided by $$ (x - x_{0}) $$, where $$ x_{0} $$ is the value of $$ x $$ for which we want to find the remainder.

Given that the polynomial is $$ f(x) = 2x^{4} - 17x^{3} + 30x^{2} + 64x + 10 $$, we can use the remainder theorem to find the remainder when $$ f(x) $$ is divided by $$ (x - x_{0}) $$ for any given value of $$ x_{0} $$.

Let's assume we want to find the remainder when $$ f(x) $$ is divided by $$ (x - 2) $$. According to the remainder theorem, the remainder is equal to $$ f(2) $$.

Substitute $$ x = 2 $$ into the polynomial $$ f(x) $$ to find the remainder:
$$ f(2) = 2(2)^{4} - 17(2)^{3} + 30(2)^{2} + 64(2) + 10 $$

Now, let's calculate the value of $$ f(2) $$ to find the remainder.
Calculate the value by following steps:
- step0: Calculate:
  $$2\times 2^{4}-17\times 2^{3}+30\times 2^{2}+64\times 2+10$$
- step1: Calculate:
  $$2^{5}-17\times 2^{3}+30\times 2^{2}+64\times 2+10$$
- step2: Multiply the terms:
  $$2^{5}-136+30\times 2^{2}+64\times 2+10$$
- step3: Multiply the terms:
  $$2^{5}-136+120+64\times 2+10$$
- step4: Multiply the numbers:
  $$2^{5}-136+120+128+10$$
- step5: Evaluate the power:
  $$32-136+120+128+10$$
- step6: Calculate:
  $$154$$
The remainder when the polynomial $$ f(x) = 2x^{4} - 17x^{3} + 30x^{2} + 64x + 10 $$ is divided by $$ (x - 2) $$ is 154.

Therefore, the value of the polynomial $$ f(x) $$ when $$ x = 2 $$ is 154.
The remainder when $$ f(x) = 2x^{4} - 17x^{3} + 30x^{2} + 64x + 10 $$ is divided by $$ (x - 2) $$ is 154.

Use the remainder theorem to evaluate the polynomial for the given values of \( x \). \[ f(x)=2 x^{4}-17 x^{3}+30 x^{2}+64 x+10 \]

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