বহুপদ - নৱম শ্ৰেণী (গণিত)

(i) @p(x) = 3x + 1, x = -\frac{1}{3}@

@ \begin{align} p(x) & = 3x + 1 \\ p(-\frac{1}{3}) & = 3(-\frac{1}{3}) + 1 \\ & = -1 + 1 \\ & = 0 \end{align} @

(ii) @p(x) = 5x - \pi, x = \frac{4}{\pi}@

@ \begin{align} p(x) &= 5x - \pi \\ p(\frac{4}{\pi}) &= 5(\frac{4}{\pi}) - \pi \\ &= \frac{20}{\pi} - \pi \\ &= \frac{20 - \pi^2}{\pi} \end{align} @

(iii) @p(x) = x^2 - 1, x = 1, -1@

@ \begin{align} p(x) &= x^2 - 1 \\ p(1) &= 1^2 - 1 = 0 \\ p(-1) &= (-1)^2 - 1 = 0 \end{align} @

(iv) @p(x) = (x + 1)(x - 2), x = -1, 2@

@ \begin{align} p(x) &= (x + 1)(x - 2) \\ p(-1) &= (-1 + 1)(-1 - 2) = 0 \\ p(2) &= (2 + 1)(2 - 2) = 0 \end{align} @

(v) @p(x) = x^2, x = 0@

@ \begin{align} p(x) &= x^2 \\ p(0) &= 0^2 = 0 \end{align} @

(vi) @p(x) = lx + m, x = -\frac{m}{l}@

@ \begin{align} p(x) &= lx + m \\ p(-\frac{m}{l}) &= l(-\frac{m}{l}) + m \\ &= -m + m \\ &= 0 \end{align} @

(vii) @p(x) = 3x^2 - 1, x = -\frac{1}{\sqrt{3}}, \frac{2}{\sqrt{3}}@

@ \begin{align} p(x) &= 3x^2 - 1 \\ p(-\frac{1}{\sqrt{3}}) &= 3(-\frac{1}{\sqrt{3}})^2 - 1 = 3(\frac{1}{3}) - 1 = 0 \\ p(\frac{2}{\sqrt{3}}) &= 3(\frac{2}{\sqrt{3}})^2 - 1 = 3(\frac{4}{3}) - 1 = 3 \end{align} @

(viii) @p(x) = 2x + 1, x = \frac{1}{2}@

@ \begin{align} p(x) &= 2x + 1 \\ p(\frac{1}{2}) &= 2(\frac{1}{2}) + 1 = 1 + 1 = 2 \end{align} @

@ \require{enclose} \begin{array}{rll} x^2 + 2x + 1 && \\[-3pt] x + 1 \enclose{longdiv}{x^3 + 3x^2 + 3x + 1}\kern-.2ex \\[-3pt] \underline{-(x^3 + x^2)} \phantom{+ 3x + 1} && \\[-3pt] 2x^2 + 3x \phantom{+ 1} && \\[-3pt] \underline{-(2x^2 + 2x)} \phantom{+ 1} && \\[-3pt] x + 1 && \\[-3pt] \underline{-(x + 1)} && \\[-3pt] 0 && \end{array} @

ভাগশেষ @0@

@ \begin{array}{rll} x^2 + \frac{7}{2}x + \frac{19}{4} && \\[-3pt] x - \frac{1}{2} \enclose{longdiv}{x^3 + 3x^2 + 3x + 1}\kern-.2ex \\[-3pt] \underline{-(x^3 - \frac{1}{2}x^2)} \phantom{+ 3x + 1} && \\[-3pt] \frac{7}{2}x^2 + 3x \phantom{+ 1} && \\[-3pt] \underline{-(\frac{7}{2}x^2 - \frac{7}{4}x)} \phantom{+ 1} && \\[-3pt] \frac{19}{4}x + 1 && \\[-3pt] \underline{-(\frac{19}{4}x - \frac{19}{8})} && \\[-3pt] \frac{27}{8} && \end{array} @

ভাগশেষ @\frac{27}{8}@

@ \begin{array}{rll} x^2 + 3x + 3 && \\[-3pt] x \enclose{longdiv}{x^3 + 3x^2 + 3x + 1}\kern-.2ex \\[-3pt] \underline{-x^3} \phantom{+ 3x^2 + 3x + 1} && \\[-3pt] 3x^2 + 3x \phantom{+ 1} && \\[-3pt] \underline{-3x^2} \phantom{+ 3x + 1} && \\[-3pt] 3x + 1 && \\[-3pt] \underline{-3x} \phantom{+ 1} && \\[-3pt] 1 && \end{array} @

ভাগশেষ @1@

@ \begin{array}{l} \phantom{x+\pi)\;x^3+{}} x^2 + (3-\pi)x + (3-3\pi+\pi^2) \\[-3pt] x + \pi \enclose{longdiv}{x^3 + 3x^2 + 3x + 1}\kern-.2ex \\[-3pt] \phantom{x+\pi)\;} \underline{-(x^3 + \pi x^2)} \\[-3pt] \phantom{x+\pi)\;x^3+{}} (3-\pi)x^2 + 3x \\[-3pt] \phantom{x+\pi)\;x^3+{}} \underline{-((3-\pi)x^2 + (3\pi-\pi^2)x)} \\[-3pt] \phantom{x+\pi)\;x^3+{}+3x^2+{}} (3-3\pi+\pi^2)x + 1 \\[-3pt] \phantom{x+\pi)\;x^3+{}+3x^2+{}} \underline{-((3-3\pi+\pi^2)x + (3\pi-3\pi^2+\pi^3))} \\[-3pt] \phantom{x+\pi)\;x^3+{}+3x^2+{}+3x+{}} 1 - 3\pi + 3\pi^2 - \pi^3 \end{array} @

ভাগশেষ @- \pi^3 + 3\pi^2 - 3\pi + 1@

@ \begin{array}{rll} \frac{1}{2}x^2 + \frac{1}{4}x + \frac{7}{8} && \\[-3pt] 2x + 5 \enclose{longdiv}{x^3 + 3x^2 + 3x + 1}\kern-.2ex \\[-3pt] \underline{-(x^3 + \frac{5}{2}x^2)} \phantom{+ 3x + 1} && \\[-3pt] \frac{1}{2}x^2 + 3x \phantom{+ 1} && \\[-3pt] \underline{-(\frac{1}{2}x^2 + \frac{5}{4}x)} \phantom{+ 1} && \\[-3pt] \frac{7}{4}x + 1 && \\[-3pt] \underline{-(\frac{7}{4}x + \frac{35}{8})} && \\[-3pt] -\frac{27}{8} && \end{array} @

ভাগশেষ @-\frac{27}{8}@

@ \begin{array}{rll} x^2 - 2x - 2 && \\[-3pt] x - 2 \enclose{longdiv}{x^3 - 4x^2 + 2x + 5}\kern-.2ex \\[-3pt] \underline{-(x^3 - 2x^2)} \phantom{+ 2x + 5} && \\[-3pt] -2x^2 + 2x \phantom{+ 5} && \\[-3pt] \underline{-(-2x^2 + 4x)} \phantom{+ 5} && \\[-3pt] -2x + 5 && \\[-3pt] \underline{-(-2x + 4)} && \\[-3pt] 1 && \end{array} @

@\therefore@ ভাগফল @x^2 - 2x - 2@ আৰু ভাগশেষ @1@

@ \begin{array}{rll} 2x - 1 && \\[-3pt] 2x^2 - 1 \enclose{longdiv}{4x^3 - 2x^2 + 0x - 3}\kern-.2ex \\[-3pt] \underline{-(4x^3 - 2x)} \phantom{- 2x^2 - 3} && \\[-3pt] -2x^2 + 2x - 3 && \\[-3pt] \underline{-(-2x^2 + 1)} \phantom{+ 2x - 3} && \\[-3pt] 2x - 4 && \end{array} @

@\therefore@ ভাগফল @2x - 1@ আৰু ভাগশেষ @2x - 4@

@ \begin{array}{rll} x^2 - 2x + 4 && \\[-3pt] 3x + 1 \enclose{longdiv}{3x^3 - 5x^2 + 10x - 3}\kern-.2ex \\[-3pt] \underline{-(3x^3 + x^2)} \phantom{+ 10x - 3} && \\[-3pt] -6x^2 + 10x \phantom{- 3} && \\[-3pt] \underline{-(-6x^2 - 2x)} \phantom{- 3} && \\[-3pt] 12x - 3 && \\[-3pt] \underline{-(12x + 4)} && \\[-3pt] -7 && \end{array} @

@\therefore@ ভাগফল @x^2 - 2x + 4@ আৰু ভাগশেষ @-7@

@ \begin{array}{rll} x^{10} - x^9 + x^8 - \dots + 1 && \\[-3pt] x + 1 \enclose{longdiv}{x^{11} + 0x^{10} + \dots - 5}\kern-.2ex \\[-3pt] \underline{-(x^{11} + x^{10})} \phantom{+ \dots - 5} && \\[-3pt] -x^{10} + 0x^9 \phantom{+ \dots - 5} && \\[-3pt] \underline{-(-x^{10} - x^9)} \phantom{+ \dots - 5} && \\[-3pt] x^9 + 0x^8 \phantom{+ \dots - 5} && \\[-3pt] \vdots && \\[-3pt] x + 1 \phantom{- 6} && \\[-3pt] \underline{-(x + 1)} && \\[-3pt] -6 && \end{array} @

@\therefore@ ভাগফল @x^{10} - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1@ আৰু ভাগশেষ @-6@

@ \begin{array}{rll} x - 4 && \\[-3pt] 3x + 10 \enclose{longdiv}{3x^2 - 2x - 40}\kern-.2ex \\[-3pt] \underline{-(3x^2 + 10x)} \phantom{- 40} && \\[-3pt] -12x - 40 && \\[-3pt] \underline{-(-12x - 40)} && \\[-3pt] 0 && \end{array} @

ভাগফল @x - 4@ আৰু ভাগশেষ @0@

প্ৰদত্ত বহুপদ ৰাশিটোৰ প্ৰমান ক্ৰম হ’ব @2x^3 + 7x^2 + 7x + 4@

@ \begin{array}{rll} x^2 + 3x + 2 && \\[-3pt] 2x + 1 \enclose{longdiv}{2x^3 + 7x^2 + 7x + 4}\kern-.2ex \\[-3pt] \underline{-(2x^3 + x^2)} \phantom{+ 7x + 4} && \\[-3pt] 6x^2 + 7x \phantom{+ 4} && \\[-3pt] \underline{-(6x^2 + 3x)} \phantom{+ 4} && \\[-3pt] 4x + 4 && \\[-3pt] \underline{-(4x + 2)} && \\[-3pt] 2 && \end{array} @

ভাগফল @x^2 + 3x + 2@ আৰু ভাগশেষ @2@

@ \begin{array}{rll} -7x + 4 && \\[-3pt] 2x + 3 \enclose{longdiv}{-14x^2 - 13x + 12}\kern-.2ex \\[-3pt] \underline{-(-14x^2 - 21x)} \phantom{+ 12} && \\[-3pt] 8x + 12 && \\[-3pt] \underline{-(8x + 12)} && \\[-3pt] 0 && \end{array} @

@\because@ ভাগশেষ @0@, গতিকে @-14x^2 - 13x + 12@ বহুপদটো @2x+3@ ৰে সম্পূৰ্ণকৈ বিভাজ্য।

@ \begin{array}{rll} x^2 + 9x + 60 && \\[-3pt] x - 7 \enclose{longdiv}{x^3 + 2x^2 - 3x + 4}\kern-.2ex \\[-3pt] \underline{-(x^3 - 7x^2)} \phantom{- 3x + 4} && \\[-3pt] 9x^2 - 3x \phantom{+ 4} && \\[-3pt] \underline{-(9x^2 - 63x)} \phantom{+ 4} && \\[-3pt] 60x + 4 && \\[-3pt] \underline{-(60x - 420)} && \\[-3pt] 424 && \end{array} @

@\because@ ভাগশেষ @424 \ne 0@, গতিকে @x - 7@ ৰাশিটো @x^3 + 2x^2 - 3x + 4@ বহুপদটোৰ এটা উৎপাদক নহয়।

(iii) @p(x) = x^4 + 3x^3 + 3x^2 + x + 1@

@ \begin{align} p(-1) &= (-1)^4 + 3(-1)^3 + 3(-1)^2 + (-1) + 1 \\ &= 1 - 3 + 3 - 1 + 1 \\ &= 1 \ne 0 \end{align} @

(iv) @p(x) = x^3 - x^2 - (2 + \sqrt{2})x + \sqrt{2}@

@ \begin{align} p(-1) &= (-1)^3 - (-1)^2 - (2 + \sqrt{2})(-1) + \sqrt{2} \\ &= -1 - 1 + 2 + \sqrt{2} + \sqrt{2} \\ &= -2 + 2 + 2\sqrt{2} \\ &= 2\sqrt{2} \ne 0 \end{align} @

(i) @p(x) = 2x^3 + x^2 - 2x - 1, g(x) = x + 1@

@g(x) = x + 1@ ৰ শূন্য হ’ল @-1@

@ \begin{align} p(-1) &= 2(-1)^3 + (-1)^2 - 2(-1) - 1 \\ &= -2 + 1 + 2 - 1 \\ &= 0 \end{align} @

যিহেতু @p(-1) = 0@, গতিকে @g(x)@, @p(x)@ ৰ এটা উৎপাদক হয়।

(ii) @p(x) = x^3 + 3x^2 + 3x + 1, g(x) = x + 2@

@g(x) = x + 2@ ৰ শূন্য হ’ল @-2@

@ \begin{align} p(-2) &= (-2)^3 + 3(-2)^2 + 3(-2) + 1 \\ &= -8 + 12 - 6 + 1 \\ &= -1 \ne 0 \end{align} @

যিহেতু @p(-2) \ne 0@, গতিকে @g(x)@, @p(x)@ ৰ এটা উৎপাদক নহয়।

(iii) @p(x) = x^3 - 4x^2 + x + 6, g(x) = x - 3@

@g(x) = x - 3@ ৰ শূন্য হ’ল @3@

@ \begin{align} p(3) &= 3^3 - 4(3)^2 + 3 + 6 \\ &= 27 - 36 + 3 + 6 \\ &= 36 - 36 \\ &= 0 \end{align} @

যিহেতু @p(3) = 0@, গতিকে @g(x)@, @p(x)@ ৰ এটা উৎপাদক হয়।

ধৰাহ’ল @p(x) = x^3 - 2x^2 - x + 2@
@2@ ৰ উৎপাদকবোৰ হ’ল @\pm 1, \pm 2@।
@x = 1@ লৈ পাওঁ,
@p(1) = 1^3 - 2(1)^2 - 1 + 2 = 1 - 2 - 1 + 2 = 0@
@\therefore (x - 1)@ ক @p(x)@ ৰ এটা উৎপাদক।

এতিয়া,
@ \begin{align} &= x^3 - 2x^2 - x + 2 \\ &= x^2(x - 1) - x(x - 1) - 2(x - 1) \\ &= (x - 1)(x^2 - x - 2) \\ &= (x - 1)(x^2 - 2x + x - 2) \\ &= (x - 1)\{x(x - 2) + 1(x - 2)\} \\ &= (x - 1)(x - 2)(x + 1) \end{align} @

ধৰাহ’ল @p(x) = x^3 - 3x^2 - 9x - 5@
@-5@ ৰ উৎপাদকবোৰ হ’ল @\pm 1, \pm 5@।
@x = -1@ লৈ পাওঁ,
@p(-1) = (-1)^3 - 3(-1)^2 - 9(-1) - 5 = -1 - 3 + 9 - 5 = 0@
@\therefore (x + 1)@ ক @p(x)@ ৰ এটা উৎপাদক।

এতিয়া,
@ \begin{align} &= x^3 - 3x^2 - 9x - 5 \\ &= x^2(x + 1) - 4x(x + 1) - 5(x + 1) \\ &= (x + 1)(x^2 - 4x - 5) \\ &= (x + 1)(x^2 - 5x + x - 5) \\ &= (x + 1)\{x(x - 5) + 1(x - 5)\} \\ &= (x + 1)(x - 5)(x + 1) \\ &= (x + 1)^2(x - 5) \end{align} @

ধৰাহ’ল @p(x) = x^3 + 13x^2 + 32x + 20@
@20@ ৰ উৎপাদকবোৰ হ’ল @\pm 1, \pm 2, \pm 4, \pm 5, \dots@
@x = -1@ লৈ পাওঁ,
@p(-1) = (-1)^3 + 13(-1)^2 + 32(-1) + 20 = -1 + 13 - 32 + 20 = 0@
@\therefore (x + 1)@ ক @p(x)@ ৰ এটা উৎপাদক।

এতিয়া,
@ \begin{align} &= x^3 + 13x^2 + 32x + 20 \\ &= x^2(x + 1) + 12x(x + 1) + 20(x + 1) \\ &= (x + 1)(x^2 + 12x + 20) \\ &= (x + 1)(x^2 + 10x + 2x + 20) \\ &= (x + 1)\{x(x + 10) + 2(x + 10)\} \\ &= (x + 1)(x + 10)(x + 2) \end{align} @

ধৰাহ’ল @p(y) = 2y^3 + y^2 - 2y - 1@
@y = 1@ লৈ পাওঁ,
@p(1) = 2(1)^3 + (1)^2 - 2(1) - 1 = 2 + 1 - 2 - 1 = 0@
@\therefore (y - 1)@ ক @p(y)@ ৰ এটা উৎপাদক।

এতিয়া,
@ \begin{align} &= 2y^3 + y^2 - 2y - 1 \\ &= 2y^2(y - 1) + 3y(y - 1) + 1(y - 1) \\ &= (y - 1)(2y^2 + 3y + 1) \\ &= (y - 1)(2y^2 + 2y + y + 1) \\ &= (y - 1)\{2y(y + 1) + 1(y + 1)\} \\ &= (y - 1)(y + 1)(2y + 1) \end{align} @

ধৰাহ’ল, @p(x) = x^3 + x^2 - x - 1@
@x = 1@ লৈ পাওঁ,
@p(1) = 1^3 + 1^2 - 1 - 1 = 1 + 1 - 1 - 1 = 0@
@\therefore (x - 1)@ ক @p(x)@ ৰ এটা উৎপাদক।

@ \begin{align} &= x^3 + x^2 - x - 1 \\ &= x^3 - x^2 + 2x^2 - 2x + x - 1 \\ &= x^2(x - 1) + 2x(x - 1) + 1(x - 1) \\ &= (x - 1)(x^2 + 2x + 1) \\ &= (x - 1)(x + 1)^2 \end{align} @

ধৰাহ’ল, @p(x) = x^3 + x^2 + x + 1@
@x = -1@ লৈ পাওঁ,
@p(-1) = (-1)^3 + (-1)^2 + (-1) + 1 = -1 + 1 - 1 + 1 = 0@
@\therefore (x + 1)@ ক @p(x)@ ৰ এটা উৎপাদক।

@ \begin{align} &= x^3 + x^2 + x + 1 \\ &= x^2(x + 1) + 1(x + 1) \\ &= (x + 1)(x^2 + 1) \end{align} @

ধৰাহ’ল @p(x) = x^3 + 3x^2 - 7x - 6@
@x = 2@ লৈ পাওঁ,
@p(2) = 2^3 + 3(2)^2 - 7(2) - 6 = 8 + 12 - 14 - 6 = 0@
@\therefore (x - 2)@ ক @p(x)@ ৰ এটা উৎপাদক।

এতিয়া,
@ \begin{align} &= x^3 + 3x^2 - 7x - 6 \\ &= x^2(x - 2) + 5x(x - 2) + 3(x - 2) \\ &= (x - 2)(x^2 + 5x + 3) \end{align} @

@\underline{\text{সমাধান:}}@

ধৰাহ’ল @f(x) = x^2 + px + q@ আৰু @g(x) = x^2 + mx + n@
@\because x + a@ দুয়োটাৰে এটা সাধাৰণ উৎপাদক, গতিকে @f(-a) = 0@ আৰু @g(-a) = 0@।

@ \begin{align} \therefore (-a)^2 + p(-a) + q &= 0 \\ \Rightarrow a^2 - ap + q &= 0 \quad \dots(1) \end{align} @

আৰু

@ \begin{align} (-a)^2 + m(-a) + n &= 0 \\ \Rightarrow a^2 - am + n &= 0 \quad \dots(2) \end{align} @

(1) আৰু (2) ৰ পৰা পাওঁ,

@ \begin{align} a^2 - ap + q &= a^2 - am + n \\ \Rightarrow -ap + am &= n - q \\ \Rightarrow a(m - p) &= n - q \\ \Rightarrow a &= \frac{n - q}{m - p} \\ \Rightarrow a &= \frac{-(q - n)}{-(p - m)} \\ \Rightarrow a &= \frac{q - n}{p - m} \end{align} @

(i) @103 \times 107@

@ \begin{align} &= 103 \times 107 \\ &= (100 + 3)(100 + 7) \\ &= (100)^2 + (3 + 7)(100) + (3 \times 7) \\ &= 10000 + 1000 + 21 \\ &= 11021 \end{align} @

(ii) @95 \times 96@

@ \begin{align} &= 95 \times 96 \\ &= (100 - 5)(100 - 4) \\ &= (100)^2 + (-5 - 4)(100) + (-5 \times -4) \\ &= 10000 - 900 + 20 \\ &= 9120 \end{align} @

(iii) @104 \times 96@

@ \begin{align} &= 104 \times 96 \\ &= (100 + 4)(100 - 4) \\ &= (100)^2 - (4)^2 \\ &= 10000 - 16 \\ &= 9984 \end{align} @

(i) @9x^2 + 6xy + y^2@

@ \begin{align} &= 9x^2 + 6xy + y^2 \\ &= (3x)^2 + 2(3x)(y) + y^2 \\ &= (3x + y)^2 \\ &= (3x + y)(3x + y) \end{align} @

(ii) @4y^2 - 4y + 1@

@ \begin{align} &= 4y^2 - 4y + 1 \\ &= (2y)^2 - 2(2y)(1) + 1^2 \\ &= (2y - 1)^2 \\ &= (2y - 1)(2y - 1) \end{align} @

(iii) @x^2 - \frac{y^2}{100}@

@ \begin{align} &= x^2 - \frac{y^2}{100} \\ &= x^2 - (\frac{y}{10})^2 \\ &= (x + \frac{y}{10})(x - \frac{y}{10}) \end{align} @

(i) @(x + 2y + 4z)^2@

@ \begin{align} &= (x + 2y + 4z)^2 \\ &= x^2 + (2y)^2 + (4z)^2 + 2(x)(2y) + 2(2y)(4z) + 2(4z)(x) \\ &= x^2 + 4y^2 + 16z^2 + 4xy + 16yz + 8zx \end{align} @

(ii) @(2x - y + z)^2@

@ \begin{align} &= (2x - y + z)^2 \\ &= (2x)^2 + (-y)^2 + z^2 + 2(2x)(-y) + 2(-y)(z) + 2(z)(2x) \\ &= 4x^2 + y^2 + z^2 - 4xy - 2yz + 4zx \end{align} @

(iii) @(-2x + 3y + 2z)^2@

@ \begin{align} &= (-2x + 3y + 2z)^2 \\ &= (-2x)^2 + (3y)^2 + (2z)^2 + 2(-2x)(3y) + 2(3y)(2z) + 2(2z)(-2x) \\ &= 4x^2 + 9y^2 + 4z^2 - 12xy + 12yz - 8zx \end{align} @

(iv) @(3a - 7b - c)^2@

@ \begin{align} &= (3a - 7b - c)^2 \\ &= (3a)^2 + (-7b)^2 + (-c)^2 + 2(3a)(-7b) + 2(-7b)(-c) + 2(-c)(3a) \\ &= 9a^2 + 49b^2 + c^2 - 42ab + 14bc - 6ca \end{align} @

(v) @(-2x + 5y - 3z)^2@

@ \begin{align} &= (-2x + 5y - 3z)^2 \\ &= (-2x)^2 + (5y)^2 + (-3z)^2 + 2(-2x)(5y) + 2(5y)(-3z) + 2(-3z)(-2x) \\ &= 4x^2 + 25y^2 + 9z^2 - 20xy - 30yz + 12zx \end{align} @

(vi) @(\frac{1}{4}a - \frac{1}{2}b + 1)^2@

@ \begin{align} &= (\frac{1}{4}a - \frac{1}{2}b + 1)^2 \\ &= (\frac{a}{4})^2 + (-\frac{b}{2})^2 + 1^2 + 2(\frac{a}{4})(-\frac{b}{2}) + 2(-\frac{b}{2})(1) + 2(1)(\frac{a}{4}) \\ &= \frac{a^2}{16} + \frac{b^2}{4} + 1 - \frac{ab}{4} - b + \frac{a}{2} \end{align} @

(i) @4x^2 + 9y^2 + 16z^2 + 12xy - 24yz - 16xz@

@ \begin{align} &= (2x)^2 + (3y)^2 + (-4z)^2 + 2(2x)(3y) + 2(3y)(-4z) + 2(-4z)(2x) \\ &= (2x + 3y - 4z)^2 \\ &= (2x + 3y - 4z)(2x + 3y - 4z) \end{align} @

(ii) @2x^2 + y^2 + 8z^2 - 2\sqrt{2}xy + 4\sqrt{2}yz - 8xz@

@ \begin{align} &= (-\sqrt{2}x)^2 + y^2 + (2\sqrt{2}z)^2 + 2(-\sqrt{2}x)(y) + 2(y)(2\sqrt{2}z) + 2(2\sqrt{2}z)(-\sqrt{2}x) \\ &= (-\sqrt{2}x + y + 2\sqrt{2}z)^2 \\ &= (-\sqrt{2}x + y + 2\sqrt{2}z)(-\sqrt{2}x + y + 2\sqrt{2}z) \end{align} @

(i) @(2x + 1)^3@

@ \begin{align} &= (2x + 1)^3 \\ &= (2x)^3 + 1^3 + 3(2x)(1)(2x + 1) \\ &= 8x^3 + 1 + 6x(2x + 1) \\ &= 8x^3 + 1 + 12x^2 + 6x \\ &= 8x^3 + 12x^2 + 6x + 1 \end{align} @

(ii) @(2a - 3b)^3@

@ \begin{align} &= (2a - 3b)^3 \\ &= (2a)^3 - (3b)^3 - 3(2a)(3b)(2a - 3b) \\ &= 8a^3 - 27b^3 - 18ab(2a - 3b) \\ &= 8a^3 - 27b^3 - 36a^2b + 54ab^2 \\ &= 8a^3 - 36a^2b + 54ab^2 - 27b^3 \end{align} @

(iii) @(\frac{3}{2}x + 1)^3@

@ \begin{align} &= (\frac{3}{2}x + 1)^3 \\ &= (\frac{3}{2}x)^3 + 1^3 + 3(\frac{3}{2}x)(1)(\frac{3}{2}x + 1) \\ &= \frac{27}{8}x^3 + 1 + \frac{9}{2}x(\frac{3}{2}x + 1) \\ &= \frac{27}{8}x^3 + 1 + \frac{27}{4}x^2 + \frac{9}{2}x \\ &= \frac{27}{8}x^3 + \frac{27}{4}x^2 + \frac{9}{2}x + 1 \end{align} @

(iv) @(x - \frac{2}{3}y)^3@

@ \begin{align} &= (x - \frac{2}{3}y)^3 \\ &= x^3 - (\frac{2}{3}y)^3 - 3(x)(\frac{2}{3}y)(x - \frac{2}{3}y) \\ &= x^3 - \frac{8}{27}y^3 - 2xy(x - \frac{2}{3}y) \\ &= x^3 - \frac{8}{27}y^3 - 2x^2y + \frac{4}{3}xy^2 \\ &= x^3 - 2x^2y + \frac{4}{3}xy^2 - \frac{8}{27}y^3 \end{align} @

(i) @(99)^3@

@ \begin{align} &= (99)^3 \\ &= (100 - 1)^3 \\ &= 100^3 - 1^3 - 3(100)(1)(100 - 1) \\ &= 1000000 - 1 - 300(99) \\ &= 1000000 - 1 - 29700 \\ &= 970299 \end{align} @

(ii) @(102)^3@

@ \begin{align} &= (102)^3 \\ &= (100 + 2)^3 \\ &= 100^3 + 2^3 + 3(100)(2)(100 + 2) \\ &= 1000000 + 8 + 600(102) \\ &= 1000000 + 8 + 61200 \\ &= 1061208 \end{align} @

(iii) @(998)^3@

@ \begin{align} &= (998)^3 \\ &= (1000 - 2)^3 \\ &= 1000^3 - 2^3 - 3(1000)(2)(1000 - 2) \\ &= 1000000000 - 8 - 6000(998) \\ &= 1000000000 - 8 - 5988000 \\ &= 994011992 \end{align} @

(i) @8a^3 + b^3 + 12a^2b + 6ab^2@

@ \begin{align} &= (2a)^3 + b^3 + 3(2a)^2(b) + 3(2a)(b)^2 \\ &= (2a + b)^3 \\ &= (2a + b)(2a + b)(2a + b) \end{align} @

(ii) @8a^3 - b^3 - 12a^2b + 6ab^2@

@ \begin{align} &= (2a)^3 - b^3 - 3(2a)^2(b) + 3(2a)(b)^2 \\ &= (2a - b)^3 \\ &= (2a - b)(2a - b)(2a - b) \end{align} @

(iii) @27 - 125a^3 - 135a + 225a^2@

@ \begin{align} &= 3^3 - (5a)^3 - 3(3)^2(5a) + 3(3)(5a)^2 \\ &= (3 - 5a)^3 \\ &= (3 - 5a)(3 - 5a)(3 - 5a) \end{align} @

(iv) @64a^3 - 27b^3 - 144a^2b + 108ab^2@

@ \begin{align} &= (4a)^3 - (3b)^3 - 3(4a)^2(3b) + 3(4a)(3b)^2 \\ &= (4a - 3b)^3 \\ &= (4a - 3b)(4a - 3b)(4a - 3b) \end{align} @

(v) @27p^3 - \frac{1}{216} - \frac{9}{2}p^2 + \frac{1}{4}p@

@ \begin{align} &= (3p)^3 - (\frac{1}{6})^3 - 3(3p)^2(\frac{1}{6}) + 3(3p)(\frac{1}{6})^2 \\ &= (3p - \frac{1}{6})^3 \\ &= (3p - \frac{1}{6})(3p - \frac{1}{6})(3p - \frac{1}{6}) \end{align} @

(i) @x^3 + y^3 = (x + y)(x^2 - xy + y^2)@

@= (x + y)(x^2 - xy + y^2)@

@ \begin{align} \text{সোঁপক্ষ} &= x(x^2 - xy + y^2) + y(x^2 - xy + y^2) \\ &= x^3 - x^2y + xy^2 + x^2y - xy^2 + y^3 \\ &= x^3 + y^3 \\ &= \text{বাওঁপক্ষ} \end{align} @

(সত‍্যাপিত হ’ল)

(ii) @x^3 - y^3 = (x - y)(x^2 + xy + y^2)@

@ \begin{align} \text{সোঁপক্ষ} &= x(x^2 + xy + y^2) - y(x^2 + xy + y^2) \\ &= x^3 + x^2y + xy^2 - x^2y - xy^2 - y^3 \\ &= x^3 - y^3 \\ &= \text{বাওঁপক্ষ} \end{align} @

(সত‍্যাপিত হ’ল)

(i) @27y^3 + 125z^3@

@ \begin{align} &= 27y^3 + 125z^3 \\ &= (3y)^3 + (5z)^3 \\ &= (3y + 5z)((3y)^2 - (3y)(5z) + (5z)^2) \\ &= (3y + 5z)(9y^2 - 15yz + 25z^2) \end{align} @

(ii) @64m^3 - 343n^3@

@ \begin{align} &= 64m^3 - 343n^3 \\ &= (4m)^3 - (7n)^3 \\ &= (4m - 7n)((4m)^2 + (4m)(7n) + (7n)^2) \\ &= (4m - 7n)(16m^2 + 28mn + 49n^2) \end{align} @

সোঁপক্ষ @= \frac{1}{2}(x + y + z)[(x - y)^2 + (y - z)^2 + (z - x)^2]@

@ \begin{align} &= \frac{1}{2}(x + y + z)[(x^2 - 2xy + y^2) + (y^2 - 2yz + z^2) + (z^2 - 2zx + x^2)] \\ &= \frac{1}{2}(x + y + z)[2x^2 + 2y^2 + 2z^2 - 2xy - 2yz - 2zx] \\ &= \frac{1}{2}(x + y + z) \times 2(x^2 + y^2 + z^2 - xy - yz - zx) \\ &= (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx) \\ &= x^3 + y^3 + z^3 - 3xyz \end{align} @

@= \text{বাওঁপক্ষ}@ (সত‍্যাপিত হ’ল)

13. (i) যদি @x + y + z = 0@, তেন্তে দেখুওৱা যে @x^3 + y^3 + z^3 = 3xyz@

(ii) যদি @a + b + c = 0@, তেন্তে দেখুওৱা যে @a^2(b+c) + b^2(c+a) + c^2(a+b) + 3abc = 0@

(iii) যদি @2a - b + c = 0@, তেন্তে দেখুওৱা যে @4a^2 - b^2 + c^2 + 4ac = 0@

(iv) যদি @a + b + c = 0@, তেন্তে দেখুওৱা যে @\frac{(b+c)^2}{bc} + \frac{(c+a)^2}{ca} + \frac{(a+b)^2}{ab} = 3@

(v) যদি @a^2 + b^2 + c^2 - ab - bc - ca = 0@, তেন্তে দেখুওৱা যে @a = b = c@

(i) যদি @x + y + z = 0@, তেন্তে দেখুওৱা যে @x^3 + y^3 + z^3 = 3xyz@

@\underline{\text{সমাধান:}}@

আমি জানো যে,

@x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)@

এতিয়া, যদি @x + y + z = 0@ হয়, তেন্তে

@ \begin{align} &= x^3 + y^3 + z^3 - 3xyz \\ &= 0 \times (x^2 + y^2 + z^2 - xy - yz - zx) \\ &\Rightarrow x^3 + y^3 + z^3 - 3xyz = 0 \\ & \Rightarrow x^3 + y^3 + z^3 = 3xyz \end{align} @

(দেখুওৱা হ’ল)

(ii) যদি @a + b + c = 0@, তেন্তে দেখুওৱা যে @a^2(b+c) + b^2(c+a) + c^2(a+b) + 3abc = 0@

@\underline{\text{সমাধান:}}@

দিয়া আছে, @a + b + c = 0@

@\Rightarrow b + c = -a, \; c + a = -b, \; a + b = -c@

@ \begin{align} \text{বাওঁপক্ষ} &= a^2(b+c) + b^2(c+a) + c^2(a+b) + 3abc \\ &= a^2(-a) + b^2(-b) + c^2(-c) + 3abc \\ &= -a^3 - b^3 - c^3 + 3abc \\ &= -(a^3 + b^3 + c^3 - 3abc) \\ &= -(0) \quad [\because a + b + c = 0] \\ &= 0 \\ &= \text{সোঁপক্ষ} \end{align} @

(দেখুওৱা হ’ল)

(iii) যদি @2a - b + c = 0@, তেন্তে দেখুওৱা যে @4a^2 - b^2 + c^2 + 4ac = 0@

@\underline{\text{সমাধান:}}@

দিয়া আছে,

@ \begin{align} & 2a - b + c = 0 \\ \Rightarrow \; & b = 2a + c \\ \Rightarrow \; & b^2 = (2a + c)^2 \\ \Rightarrow \; & b^2 = 4a^2 + 4ac + c^2 \\ \Rightarrow \; & 4a^2 - b^2 + c^2 + 4ac = 0 \end{align} @

(দেখুওৱা হ’ল)

(iv) যদি @a + b + c = 0@, তেন্তে দেখুওৱা যে @\frac{(b+c)^2}{bc} + \frac{(c+a)^2}{ca} + \frac{(a+b)^2}{ab} = 3@

@\underline{\text{সমাধান:}}@

দিয়া আছে,

@a + b + c = 0@
@\therefore@ @b + c = -a@, @c + a = -b@, @a + b = -c@

@ \begin{align} \text{বাওঁপক্ষ} &= \frac{(b+c)^2}{bc} + \frac{(c+a)^2}{ca} + \frac{(a+b)^2}{ab} \\ &= \frac{(-a)^2}{bc} + \frac{(-b)^2}{ca} + \frac{(-c)^2}{ab} \\ &= \frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab} \\ &= \frac{a^3 + b^3 + c^3}{abc} \\ &= \frac{3abc}{abc} \quad [\because a + b + c = 0] \\ &= 3 \\ &= \text{সোঁপক্ষ} \end{align} @

(দেখুওৱা হ’ল)

(v) যদি @a^2 + b^2 + c^2 - ab - bc - ca = 0@, তেন্তে দেখুওৱা যে @a = b = c@

@\underline{\text{সমাধান:}}@

দিয়া আছে, @a^2 + b^2 + c^2 - ab - bc - ca = 0@

দুয়োপক্ষক 2 ৰে পূৰণ কৰি পাওঁ,

@ \begin{align} & 2(a^2 + b^2 + c^2 - ab - bc - ca) = 2 \times 0 \\ \Rightarrow \; & 2a^2 + 2b^2 + 2c^2 - 2ab - 2bc - 2ca = 0 \\ \Rightarrow \; & (a^2 - 2ab + b^2) + (b^2 - 2bc + c^2) + (c^2 - 2ca + a^2) = 0 \\ \Rightarrow \; & (a - b)^2 + (b - c)^2 + (c - a)^2 = 0 \end{align} @

যিহেতু বৰ্গ ৰাশিৰ সমষ্টি শূণ্য, গতিকে প্ৰতিটো ৰাশিৰ মান শূণ্য হ’ব।

@ \begin{align} \therefore & \; (a - b)^2 = 0 \\ & \Rightarrow a - b = 0 \\ & \Rightarrow a = b \end{align} @

ঠিক একেদৰে,

@b = c@ আৰু @c = a@

@\therefore a = b = c@

(দেখুওৱা হ’ল)

@ \begin{align} & 25a^2 - 35a + 12 \\ &= 25a^2 - 20a - 15a + 12 \\ &= 5a(5a - 4) - 3(5a - 4) \\ &= (5a - 4)(5a - 3) \end{align} @

@\therefore@ সম্ভাব্য দীঘ আৰু প্ৰস্থ হ’ল @(5a - 3)@ আৰু @(5a - 4)@

@ \begin{align} & 35y^2 + 13y - 12 \\ &= 35y^2 + 28y - 15y - 12 \\ &= 7y(5y + 4) - 3(5y + 4) \\ &= (5y + 4)(7y - 3) \end{align} @

@\therefore@ সম্ভাব্য দীঘ আৰু প্ৰস্থ হ’ল @(7y - 3)@ আৰু @(5y + 4)@

@ \begin{align} & 3x^2 - 12x \\ &= 3x(x - 4) \end{align} @

যিহেতু আয়তন = দীঘ @\times@ প্ৰস্থ @\times@ উচ্চতা

@\therefore@ সম্ভাব্য মাত্ৰাকেইটা হ’ল @3, x@ আৰু @(x - 4)@

@ \begin{align} & 12ky^2 + 8ky - 20k \\ &= 4k(3y^2 + 2y - 5) \\ &= 4k(3y^2 + 5y - 3y - 5) \\ &= 4k \{y(3y + 5) - 1(3y + 5)\} \\ &= 4k(3y + 5)(y - 1) \end{align} @

@\therefore@ সম্ভাব্য মাত্ৰাকেইটা হ’ল @4k, (3y + 5)@ আৰু @(y - 1)@