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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.

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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