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Question Number 32681 by rahul 19 last updated on 31/Mar/18

Total no. of polynomials of the form x^3 +ax^2 +bx+c that are divisible by x^2 +1, where a,b,c∈1,2,3,....,10 is 1) 10 2) 15 3) 5 4) 8

$$\boldsymbol{{T}}{otal}\:{no}.\:{of}\:{polynomials}\:{of}\:{the}\:{form} \\ $$$${x}^{\mathrm{3}} +{ax}^{\mathrm{2}} +{bx}+{c}\:\:{that}\:{are}\:{divisible}\:{by}\: \\ $$$${x}^{\mathrm{2}} +\mathrm{1},\:{where}\:{a},{b},{c}\in\mathrm{1},\mathrm{2},\mathrm{3},....,\mathrm{10}\:{is}\: \\ $$$$\left.\mathrm{1}\right)\:\mathrm{10} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{15} \\ $$$$\left.\mathrm{3}\right)\:\mathrm{5} \\ $$$$\left.\mathrm{4}\right)\:\mathrm{8} \\ $$

Commented by Rasheed.Sindhi last updated on 31/Mar/18

If x^2 +1 is one factor of x^3 +ax^2 +bx+c then other factor will be linear and of type x+p. (x^2 +1)(x+p)=x^3 +px^2 +x+p Comparing with given polynomial we have b=1& a=c=p Hence the given polynomial will be of the form: x^3 +ax^2 +x+a Since a ∈1,2,3,...,10 (Ten values) Hence total number of polynomials of the given form with given conditions is 10

$$\mathrm{If}\:{x}^{\mathrm{2}} +\mathrm{1}\:\mathrm{is}\:\mathrm{one}\:\mathrm{factor}\:\mathrm{of}\:{x}^{\mathrm{3}} +{ax}^{\mathrm{2}} +{bx}+{c} \\ $$$$\mathrm{then}\:\mathrm{other}\:\mathrm{factor}\:\mathrm{will}\:\mathrm{be}\:\mathrm{linear}\:\mathrm{and}\:\mathrm{of} \\ $$$$\mathrm{type}\:{x}+{p}. \\ $$$$\:\:\left({x}^{\mathrm{2}} +\mathrm{1}\right)\left({x}+{p}\right)={x}^{\mathrm{3}} +{px}^{\mathrm{2}} +{x}+{p} \\ $$$$\mathrm{Comparing}\:\mathrm{with}\:\mathrm{given}\:\mathrm{polynomial} \\ $$$$\mathrm{we}\:\mathrm{have}\:{b}=\mathrm{1\&}\:{a}={c}={p} \\ $$$$\mathrm{Hence}\:\mathrm{the}\:\mathrm{given}\:\mathrm{polynomial}\:\mathrm{will}\:\mathrm{be} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{form}: \\ $$$$\:\:\:\:\:\:{x}^{\mathrm{3}} +{ax}^{\mathrm{2}} +{x}+{a} \\ $$$$\mathrm{Since}\:{a}\:\in\mathrm{1},\mathrm{2},\mathrm{3},...,\mathrm{10}\:\left(\mathrm{Ten}\:\mathrm{values}\right) \\ $$$$\mathrm{Hence}\:\mathrm{total}\:\mathrm{number}\:\mathrm{of}\:\mathrm{polynomials} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{given}\:\mathrm{form}\:\mathrm{with}\:\mathrm{given}\:\mathrm{conditions} \\ $$$$\mathrm{is}\:\:\mathrm{10} \\ $$

Commented by rahul 19 last updated on 31/Mar/18

thank u sir!

$${thank}\:{u}\:{sir}! \\ $$

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