Find the value of k such that `(x^3-k*x^2+2)` / `(x-1)` has a remainder of 8.

Thursday, October 13, 2011

Find the value of k such that $\displaystyle{\left({{x}}^{{3}}-{k}\cdot{{x}}^{{2}}+{2}\right)}$ / $\displaystyle{\left({x}-{1}\right)}$ has a remainder of 8.

Hello!

There is the Polynomial remainder theorem (its prove is simple provided we know that the remainder is a number (a monomial of degree 0)):

if a polynomial $\displaystyle{P}{\left({x}\right)}$ is divided by a binomial $\displaystyle{\left({x}-{c}\right)}$ then the remainder is equal to $\displaystyle{P}{\left({c}\right)}.$

Here $\displaystyle{P}{\left({x}\right)}={{x}}^{{3}}-{k}\cdot{{x}}^{{2}}+{2}$ and $\displaystyle{c}={1},$ therefore the remainder is equal to

$\displaystyle{P}{\left({1}\right)}={1}-{k}+{2}={3}-{k}.$

And it is equal to 8 by the condition of the problem,

$\displaystyle{3}-{k}={8}.$

Therefore k=3-8=-5. This is the (unique) answer, k=-5.

Check the answer by the direct division:

$\displaystyle\frac{{{{x}}^{{3}}+{5}{{x}}^{{2}}+{2}}}{{{x}-{1}}}=\frac{{{{x}}^{{2}}{\left({x}-{1}\right)}+{{x}}^{{2}}+{5}{{x}}^{{2}}+{2}}}{{{x}-{1}}}=$

$\displaystyle={{x}}^{{2}}+\frac{{{6}{{x}}^{{2}}+{2}}}{{{x}-{1}}}={{x}}^{{2}}+\frac{{{6}{x}{\left({x}-{1}\right)}+{6}{x}+{2}}}{{{x}-{1}}}=$

$\displaystyle={{x}}^{{2}}+{6}{x}+\frac{{{6}{x}+{2}}}{{{x}-{1}}}={{x}}^{{2}}+{6}{x}+\frac{{{6}{\left({x}-{1}\right)}+{6}+{2}}}{{{x}-{1}}}=$

$\displaystyle={\left({{x}}^{{2}}+{6}{x}+{6}\right)}+\frac{{8}}{{{x}-{1}}},$

$\displaystyle{\left({{x}}^{{3}}+{5}{{x}}^{{2}}+{2}\right)}={\left({{x}}^{{2}}+{6}{x}+{6}\right)}{\left({x}-{1}\right)}+{8}.$

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Thursday, October 13, 2011

Find the value of k such that $\displaystyle{\left({{x}}^{{3}}-{k}\cdot{{x}}^{{2}}+{2}\right)}$ / $\displaystyle{\left({x}-{1}\right)}$ has a remainder of 8.

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What was the device called which Faber had given Montag in order to communicate with him?

Thursday, October 13, 2011

Find the value of k such that / has a remainder of 8.

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Post a Comment

What was the device called which Faber had given Montag in order to communicate with him?

Find the value of k such that $\displaystyle{\left({{x}}^{{3}}-{k}\cdot{{x}}^{{2}}+{2}\right)}$ / $\displaystyle{\left({x}-{1}\right)}$ has a remainder of 8.