Algebraic Manipulation

Algebraic Manipulation - Substitution


If x12+x12=10,x^{\frac{1}{2}}+x^{-\frac{1}{2}}=10, what is the value of x+x1x+x^{-1}?

Consider the polynomial f(x)=x3+ax+bf(x)=x^3+ax+b, where aa and bb are constants. If f(x+1004)f(x+1004) leaves a remainder of 5252 upon division by x+1005x+1005, and f(x+1005)f(x+1005) leaves a remainder of 66 upon division by x+1004x+1004, what is the value of a+ba+b?

The polynomial xn(x2+ax+b)x^n (x^2+ax+b) leaves a remainder of 9n(x9)9 ^n (x-9) upon division by (x9)2,(x-9)^2, where aa and bb are constants and n2n\geq 2 is an integer. What is the value of bab-a?

If a=b+4 a = b + 4 , what is the value of

3a3b+3? 3 a - 3 b + 3 ?

If the equality (x22x+6)3=a6x6+a5x5++a1x1+a0(x^2-2x+6)^3=a_6 x^6+a_5 x^5+\cdots+a_1 x^1+a_0 always holds regardless the value of xx, what is a0+a2+a4+a6?a_0+a_2+a_4+a_6?


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