# Number Theory (Polynomials) Find The Remainder?

A polynomial $$f(x) = x^{50} + 3x^{49} + 3x + 12$$ when divided by $$x - a$$, it leaves remainder $$3$$ & when its quotient is further divided by $$x - b$$ it leaves remainder $$5$$, also when $$f(x)$$ is divided by $$x^2 - ( b + a)x + ab$$ the remainder is $$x + 6$$. Find $$a$$?

• The Answer "a" might come in terms of x as well
– Anuj
Oct 30 '19 at 13:50
• Where this polynomial belong to? You are considering integer coefficients or complex? Oct 31 '19 at 10:00

So by Remainder theorem, $$f(a)=3$$ and $$f(x)=g(x)(x-a)(x-b)+(x+6)$$ for some polynomial $$g(x)$$. Using the first result in the second, you immediately get …

• So, by combining the above two answers we get the value of $a$ free from $x$. Oct 31 '19 at 10:12

$$P=x^{50}+3x^{49}+3x+12=x^{49}(x+3)+3(x+3)+3=k(x-a)+3$$

So the quotient k is:

$$k=\frac{(x^{49}+3)(x+3)}{x-a}$$

$$\frac{(x^{49}+3)(x+3)}{x-a}=k_1(x-b)+5$$

$$(x^{49}+3)(x+3)=k_1(x-a)(x-b)+5(x-a)$$

$$p=(x^{49}+3)(x+3)+3=k_1(x-a)(x-b)+5(x-a)+3$$

Therefore:

$$5(x-a)+3=x+6$$$$a=\frac{4x-3}{5}$$