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# Algebraic Manipulation

Gauss, Ramanujan, and a pantheon of other mathematicians have given us algebraic manipulation tools way beyond what's taught in school. Learn what they knew.

# Algebraic Manipulation - Substitution

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

Consider the polynomial $$f(x)=x^3+ax+b$$, where $$a$$ and $$b$$ are constants. If $$f(x+1004)$$ leaves a remainder of $$52$$ upon division by $$x+1005$$, and $$f(x+1005)$$ leaves a remainder of $$6$$ upon division by $$x+1004$$, what is the value of $$a+b$$?

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

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

$3 a - 3 b + 3 ?$

If the equality $(x^2-2x+6)^3=a_6 x^6+a_5 x^5+\cdots+a_1 x^1+a_0$ always holds regardless the value of $$x$$, what is $a_0+a_2+a_4+a_6?$

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