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$$x^2 + y^2 = 25$$ $$x^2 - y^2 = 7$$
Find the remainder when $x^100 + 2x^50 + 1$ is divided by $x^2 - 1$. me las vas a pagar mary rojas pdf %C3%A1lgebra
Rewrite $4^x = (2^2)^x = (2^x)^2$ and $2^x+1 = 2 \cdot 2^x$. Let $t = 2^x$. Equation: $t^2 + 2t - 3 = 0$. Roots: $(t+3)(t-1)=0 \rightarrow t = -3$ (invalid, since $t > 0$) or $t = 1$. Thus $2^x = 1 \rightarrow x = 0$. 3. Logarithmic Revenge (Change of Base) Logarithms are where students cry. Mary Rojas’ PDF often includes nested logs. Copy the 10 exercises above onto a Word
When dividing by $x^2 - 1$, the remainder is of the form $ax + b$. We know $x^2 = 1$, so $x^100 = (x^2)^50 = 1^50 = 1$. And $x^50 = (x^2)^25 = 1$. Thus $P(x) \equiv 1 + 2(1) + 1 = 4$. Since the remainder is a constant, $ax+b = 4$. Answer: $4$ (remainder is $0\cdot x + 4$). 7. Age Problems (Verbal Algebra) Classic word problem: Rewrite $4^x = (2^2)^x = (2^x)^2$ and $2^x+1 = 2 \cdot 2^x$