Using the given zero, find one other zero of f(x). i is a zero of f(x).= x4 - 2x3 + 38x2 - 2x + 37.

1 answer:

4 0

If i is a zero, -i is also a zero.
The way you get a complex zero is by taking the square root of a negative number. Taking the sqrt (-1) = + and - i

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.<br> Evaluate the series 1 + 0.1 + 0.01 + . . .

EastWind [94]

We can employ a simple repeated decimal trick:

$x=1.111\ldots$

$0.1x=0.111\ldots$

$\implies x-0.1x=1\implies0.9x=1\implies x=\dfrac1{0.9}=\dfrac{10}9$

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Alternatively, we can compute the partial sum of the series.

$\displaystyle S_n=\sum_{k=0}^n\dfrac1{10^k}$

$S_n=1+0.1+0.01+\cdots+\dfrac1{10^n}$

$0.1S_n=0.1+0.01+0.001+\cdots+\dfrac1{10^{n+1}}$

$\implies S_n-0.1S_n=0.9S_n=1-\dfrac1{10^{n+1}}$

$\implies S_n=\dfrac{10}9-\dfrac9{10^n}$

As $n\to\infty$ , the second term vanishes and we're left with $\dfrac{10}9$ . Notice that this is really just a more formal version of the earlier trick.

6 0

2 years ago

Read 2 more answers

Divide line segments

ser-zykov [4K]

tis noteworthy that the segment contains endpoints of A and C and the point B is in between A and C cutting the segment in a 1:2 ratio,

$\bf \textit{internal division of a line segment using ratios} \\\\\\ A(-9,-7)\qquad C(x,y)\qquad \qquad \stackrel{\textit{ratio from A to C}}{1:2} \\\\\\ \cfrac{A\underline{B}}{\underline{B} C} = \cfrac{1}{2}\implies \cfrac{A}{C}=\cfrac{1}{2}\implies 2A=1C\implies 2(-9,-7)=1(x,y)\\\\[-0.35em] ~\dotfill\\\\ B=\left(\frac{\textit{sum of "x" values}}{\textit{sum of ratios}}\quad ,\quad \frac{\textit{sum of "y" values}}{\textit{sum of ratios}}\right)\\\\[-0.35em] ~\dotfill$

$\bf B=\left(\cfrac{(2\cdot -9)+(1\cdot x)}{1+2}\quad ,\quad \cfrac{(2\cdot -7)+(1\cdot y)}{1+2}\right)~~=~~(-4,-6) \\\\[-0.35em] ~\dotfill\\\\ \cfrac{(2\cdot -9)+(1\cdot x)}{1+2}=-4\implies \cfrac{-18+x}{3}=-4 \\\\\\ -18+x=-12\implies \boxed{x=6} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{(2\cdot -7)+(1\cdot y)}{1+2}=-6\implies \cfrac{-14+y}{3}=-6 \\\\\\ -14+y=-18\implies \boxed{y=-4}$