৯ম – ১০ম শ্রেণির অনুশীলনী-৩.৪ উৎপাদকে বিশ্লেষণ (ভাগশেষ উপপাদ্য)
\begin{align*}
&\mathbf{১. 3a^{3}+2a+5}\\
&সমাধানঃ\\
&মনে \text{ } করি, f\left(a\right)=3a^{3}+2a+5\\
&f\left(-1\right)=3{\left(-1\right)}^{3}+2\left(-1\right)+5\\
&= 3.\left(-1\right)-2+5\\
&=-3-2+5\\
&=0\\
&অতএব, f(-1)=0 \text{ } হওয়ায় (a+1) হলো\text{ } f(a)-এর \text{ } একটি \text{ } উৎপাদক।\\
&3a^{3}+2a+5\\
&=3a^{3}+3a^{2}-3a^{2}-3a+5a+5\\
&=3a^{2}\left(a+1\right)-3a\left(a+1\right)+5\left(a+1\right)\\
&=\left(a+1\right)\left(3a^{2}-3a+5\right) [কমন]\\
&\mathbf{২. x^{3}-7xy^{2}-6y^{3}}\\
&সমাধানঃ\\
&মনে\text{ } করি, f\left(x\right)=x^{3}-7xy^{2}-6y^{3}\\
&f\left(-y\right)={\left(-y\right)}^{3}-7\left(-y\right)y^{2}-6y^{3}\\
&=-y^{3}+7y^{3}-6y^{3}\\
&=0\\
&অতএব, f(-y)=0 \text{ } হওয়ায় (x+y) হলো\text{ } f(x)-এর \text{ } একটি \text{ } উৎপাদক।\\
&x^{3}-7xy^{2}-6y^{3}\\
&=x^{3}-x^{2}y-x^{2}y-xy^{2}-6xy^{2}-6y^{3}\\
&=x^{2}\left(x+y\right)-xy\left(x+y\right)-6y^{2}\left(x+y\right)\\
&=\left(x+y\right)\left(x^{2}-xy-6y^{2}\right) [কমন]\\
&=\left(x+y\right)\left(x^{2}-3xy+2xy-6y^{2}\right) [ মিডল টার্ম ব্রেক ]\\
&=\left(x+y\right)\{x\left(x-3y\right)+2y\left(x-3y\right)\} [কমন]\\
&= \left(x+y\right)\left(x-3y\right)\left(x+2y\right) [কমন]\\
&=\left(x+y\right)\left(x+2y\right)\left(x-3y\right)
\end{align*}
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\begin{align*}
&\mathbf{৩. x^{3}+2x^{2}-5x-6}\\
&সমাধানঃ\\
&মনে \text{ } করি, f\left(x\right)=x^{3}+2x^{2}-5x-6\\
&f\left(-1\right)={\left(-1\right)}^{3}+2{\left(-1\right)}^{2}-5\left(-1\right)-6\\
&=-1+2.1+5-6\\
&=-1+2+5-6\\
&=0\\
&অতএব, (x+1) হলো\text{ } f(x)-এর \text{ } একটি \text{ } উৎপাদক। \\
&x^{3}+2x^{2}-5x-6\\
&=x^{3}+3x^{2}-x^{2}-3x-2x-6\\
&=x^{2}\left(x+3\right)-x\left(x+3\right)-2\left(x+3\right)\\
&=\left(x+3\right)\left(x^{2}-x-2\right) [কমন]\\
&= \ \ \left(x+3\right)\left(x^{2}-2x+x-2\right) [ মিডল টার্ম ব্রেক ]\\
&= \left(x+3\right)\{x\left(x-2\right)+1\left(x-2\right)\} [কমন]\\
&=\left(x+3\right)\left(x-2\right)\left(x+1\right) [কমন]\\
&\mathbf{৪. x^{3}+4x^{2}+x-6}\\
&সমাধানঃ\\
&মনে\text{ } করি, f\left(x\right)=x^{3}+4x^{2}+x-6\\
&f\left(1\right) = 1^{3}+{4.1}^{2}+1-6\\
&=1+4+1-6\\
&=0\\
&অতএব, (x-1) হলো\text{ } f(x)-এর \text{ } একটি \text{ } উৎপাদক।\\
&x^{3}+4x^{2}+x-6\\
&=x^{3}+3x^{2}+x^{2}+3x-2x-6\\
&=x^{2}\left(x+3\right)+x\left(x+3\right)-2\left(x+3\right)\\
&=\left(x+3\right)\left(x^{2}+x-2\right) [কমন]\\
&= \left(x+3\right)\left(x^{2}+2x-x-2\right) [ মিডল টার্ম ব্রেক ]\\
&= \left(x+3\right)\{x\left(x+2\right)-1\left(x+2\right)\} [কমন]\\
&=\left(x+3\right)\left(x+2\right)\left(x-1\right) [কমন]\\
&\mathbf{৫. a^{3}+3a+36}\\
&সমাধানঃ\\
&মনে \text{ } করি, f\left(a\right)=a^{3}+3a+36\\
&f\left(-3\right)={\left(-3\right)}^{3}+3\left(-3\right)+36\\
&=-27-9+36\\
&=0\\
&অতএব, (a+3) হলো\text{ } f\left(a\right)-এর \text{ } একটি\text{ } উৎপাদক।\\
&a^{3}+3a+36\\
&=a^{3}+3a^{2}-3a^{2}-9a+12a+36\\
&=a^{2}\left(a+3\right)-3a\left(a+3\right)+12\left(a+3\right)\\
&=\left(a+3\right)\left(a^{2}-3a+12\right) [কমন]\\
&\mathbf{৬. a^{4}-4a+3}\\
&সমাধানঃ\\
&মনে\text{ } করি, f\left(a\right)=a^{4}-4a+3\\
&f\left(1\right) = 1^{4}-4.1+3\\
&= 1-4+3\\
&= 4-4\\
&= 0\\
&অতএব, (a-1) হলো \text{ } f\left(a\right)-এর \text{ } একটি\text{ } উৎপাদক।\\
&a^{4}-4a+3\\
&= a^{4}-a^{3}+a^{3}-a^{2}+a^{2}-a-3a+3\\
&= a^{3}\left(a-1\right)+a^{2}\left(a-1\right)+a\left(a-1\right)-3\left(a-1\right)\\
&=\left(a-1\right)\left(a^{3}+a^{2}+a-3\right) [কমন]\\
&অবশিষ্ট রাশি a^{3}+a^{2}+a-3\\
&এখন, \\
&f\left(a\right)=\ a^{3}+a^{2}+a-3\\
&f\left(1\right) = {\ 1}^{3}+1^{2}+1-3\\
&= 1+1+1-3\\
&= 3-3\\
&= 0\\
&অতএব, (a-1) হলো\text{ } f\left(a\right)-এর \text{ } একটি \text{ } উৎপাদক।\\
&তাহলে, \\
&a^{3}+a^{2}+a-3\\
&=a^{3}-a^{2}+2a^{2}-2a+3a-3\\
&=a^{2}\left(a-1\right)+2a\left(a-1\right)+3\left(a-1\right)\\
&= \left(a-1\right)\left(a^{2}+2a+3\right) [কমন]\\
&সুতরাং \\
&a^{4}-4a+3\\
&= \left(a-1\right)\left(a^{3}+a^{2}+a-3\right)\\
&= \left(a-1\right)\left(a-1\right)\left(a^{2}+2a+3\right) [মান \text{ } বসিয়ে ] \\
&\mathbf{৭. a^{3}-a^{2}-10a-8}\\
&সমাধানঃ\\
&মনে \text{ } করি, f\left(a\right)=a^{3}-a^{2}-10a-8\\
&f\left(4\right)=4^{3}-4^{2}-10\left(4\right)-8\\
&=64-16-40-8\\
&=0\\
&অতএব, (a-4) হলো\text{ } f\left(a\right)-এর\text{ } একটি \text{ } উৎপাদক।\\
&a^{3}-a^{2}-10a-8\\
&=a^{3}-4a^{2}+3a^{2}-12a+2a-8\\
&=a^{2}\left(a-4\right)+3a\left(a-4\right)+2\left(a-4\right)\\
&=\left(a-4\right)\left(a^{2}+3a+2\right) [কমন]\\
&= \left(a-4\right)\left(a^{2}+2a+a+2\right) [ মিডল টার্ম ব্রেক ]\\
&= \left(a-4\right)\{a\left(a+2\right)+1\left(a+2\right)\} [কমন]\\
&= \left(a-4\right)\left(a+2\right)\left(a+1\right) [কমন]\\
&=\left(a-4\right)\left(a+1\right)\left(a+2\right)\\
&\mathbf{৮. x^{3}-3x^{2}+4x-4}\\
&সমাধানঃ\\
&মনে\text{ } করি, f\left(x\right)=x^{3}-3x^{2}+4x-4\\
&f\left(2\right)= 2^{3}-{3.2}^{2}+4.2-4\\
&= 8-12+8-4\\
&=0\\
&অতএব, (x-2) হলো\text{ } f(x)-এর \text{ } একটি\text{ } উৎপাদক।\\
&x^{3}-3x^{2}+4x-4\\
&= x^{3}-2x^{2}-x^{2}+2x+2x-4\\
&= x^{2}\left(x-2\right)-x\left(x-2\right)+2\left(x-2\right)\\
&= \left(x-2\right)\left(x^{2}-x+2\right) [কমন]\\
&\mathbf{৯. a^{3}-7a^{2}b+7ab^{2}-b^{3}}\\
&সমাধানঃ\\
&মনে \text{ } করি, f\left(a\right)=a^{3}-7a^{2}b+7ab^{2}-b^{3}\\
&f\left(b\right) = b^{3}-7.b^{2}.b+7.b.b^{2}-b^{3}\\
&=b^{3}-7b^{3}+7b^{3}-b^{3}\\
&=0\\
&অতএব, (a-b) হলো\text{ } f\left(a\right)-এর\ একটি \text{ } উৎপাদক।\\
&a^{3}-7a^{2}b+7ab^{2}-b^{3}\\
&= a^{3}-a^{2}b-6a^{2}b+6ab^{2}+ab^{2}-b^{3}\\
&= a^{2}\left(a-b\right)-6ab\left(a-b\right)+b^{2}\left(a-b\right)\\
&=\left(a-b\right)\left(a^{2}-6ab+b^{2}\right) [কমন]\\
&\mathbf{১০. x^{3}-x-24}\\
&সমাধানঃ\\
&মনে\text{ } করি, f\left(x\right)=x^{3}-x-24\\
&f\left(3\right)= 3^{3}-3-14\\
&= 27-3-24\\
&=0\\
&অতএব, (x-3) হলো\text{ } f(x)-এর\text{ } একটি \text{ } উৎপাদক।\\
&x^{3}-x-24\\
&= x^{3}-3x^{2}+3x^{2}-9x+8x-24\\
&= x^{2}\left(x-3\right)+3x\left(x-3\right)+8\left(x-3\right)\\
&=\left(x-3\right)\left(x^{2}+3x+8\right) [কমন]\\
&\mathbf{১১. x^{3}+6x^{2}+11x+6}\\
&সমাধানঃ\\
&মনে \text{ } করি, f\left(x\right)=x^{3}+6x^{2}+11x+6\\
&f\left(-1\right) = {\left(-1\right)}^{3}+6.{\left(-1\right)}^{2}+11.\left(-1\right)+6\\
&= -1+6.1-11+6\\
&=-1+6-11+6\\
&=0\\
&অতএব, (x+1) হলো\text{ } f(x)-এর\text{ } একটি \text{ } উৎপাদক।\\
&x^{3}+6x^{2}+11x+6\\
&=x^{3}+2x^{2}+4x^{2}+8x+3x+6\\
&=x^{2}\left(x+2\right)+4x\left(x+2\right)+3\left(x+2\right)\\
&=\left(x+2\right)\left(x^{2}+4x+3\right) [কমন]\\
&= \left(x+2\right)\left(x^{2}+3x+x+3\right) [ মিডল টার্ম ব্রেক ]\\
&= \left(x+2\right)\{x\left(x+3\right)+1\left(x+3\right)\} [কমন]\\
&= \left(x+2\right)\left(x+3\right)\left(x+1\right) [কমন]\\
&=\left(x+1\right)\left(x+2\right)\left(x+3\right)\\
&\mathbf{১২. 2x^{4}-3x^{3}-3x-2}\\
&সমাধানঃ\\
&মনে \text{ } করি, f\left(x\right)=2x^{4}-3x^{3}-3x-2\\
&f\left(2\right)=2\left(2^{4}\right)-3\left(2^{3}\right)-3\left(2\right)-2\\
&= 2.16-3.8-6-2\\
&=32-24-6-2\\
&=0\\
&অতএব, (x-2) হলো\text{ } f(x)-এর\text{ } একটি \text{ } উৎপাদক।\\
&2x^{4}-3x^{3}-3x-2\\
&=2x^{4}-4x^{3}+x^{3}-2x^{2}+2x^{2}-4x+x-2\\
&=x^{3}\left(2x-4\right)+x^{2}\left(x-2\right)+2x\left(x-2\right)+\left(x-2\right)\\
&=\left(x-2\right)\left(2x^{3}+x^{2}+2x+1\right) [কমন]\\
&=\left(x-2\right)\left\{x^{2}\left(2x+1\right)+1\left(2x+1\right)\right\} [কমন]\\
&=\left(x-2\right)\left(2x+1\right)\left(x^{2}+1\right) [কমন]\\
&\mathbf{১৩. 4x^{4}+12x^{3}+7x^{2}-3x-2}\\
&সমাধানঃ\\
&মনে \text{ } করি, f\left(x\right)=4x^{4}+12x^{3}+7x^{2}-3x-2\\
&f\left(\frac{1}{2}\right)=4{\left(\frac{1}{2}\right)}^{4}+12{\left(\frac{1}{2}\right)}^{3}+7{\left(\frac{1}{2}\right)}^{2}-3\left(\frac{1}{2}\right)-2\\
&= \ \ 4.\frac{1}{16}+12.\frac{1}{8}+7.\frac{1}{4}-\frac{3}{2}-2 \\
&= \frac{1}{4}+\frac{3}{2}+\frac{7}{4}-\frac{3}{2}-2\\
&= \frac{1}{4}+\frac{7}{4}-2\\
&= \frac{1+7-4}{4}\\
&= \frac{0}{4}\\
&=0 \\
&অতএব, (2x-1) হলো \text{ } f(x)-এর \text{ } একটি \text{ } উৎপাদক।\\
&4x^{4}+12x^{3}+7x^{2}-3x-2\\
&=4x^{4}+12x^{3}+8x^{2}-x^{2}-3x-2\\
&=4x^{2}\left(x^{2}+3x+2\right)-\left(x^{2}+3x+2\right)\\
&=\left(4x^{2}-1\right)\left(x^{2}+3x+2\right) [কমন]\\
&=\ \ \{{\left(2x\right)}^{2}-1^{2}\}\left(x^{2}+2x+x+2\right) [ মিডল টার্ম ব্রেক ]\\
&=\ \left(2x+1\right)\left(2x-1\right)\{x\left(x+2\right)+1\left(x+2\right)\} \left[সূত্র\ {\ \ a}^{2}-b^{2}=\left(a+b\right)\left(a-b\right)\right]\ এবং [কমন]\\
&=\left(2x+1\right)\left(2x-1\right)\left(x+2\right)\left(x+1\right) [কমন]\\
&\mathbf{১৪. x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x}\\
&সমাধানঃ\\
&মনে \text{ } করি, f\left(x\right)=x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x\\
&f\left(1\right)= 1^{6}-1^{5}+1^{4}-1^{3}+1^{2}-1\\
&=1-1+1-1+1-1\\
&=0\\
&অতএব, (x-1) হলো\text{ } f(x)-এর\text{ } একটি\text{ } উৎপাদক।\\
&x^{6}-x^{5}+x^{4}-x^{3}+x^{2}-x\\
&=x^{5}\left(x-1\right)+x^{3}\left(x-1\right)+x\left(x-1\right)\\
&=\left(x-1\right)\left(x^{5}+x^{3}+x\right) [কমন]\\
&= \left(x-1\right).x\left(x^{4}+x^{2}+1\right) [কমন]\\
&=x\left(x-1\right)\left(x^{4}+x^{2}+1\right)\\
&= x\left(x-1\right)\left(x^{4}+2x^{2}+1-x^{2}\right)\\
&= x\left(x-1\right)\{{\left(x^{2}\right)}^{2}+2.x^{2}.1+1^{2}-x^{2}\}\\
&=x\left(x-1\right)\left\{{\left(x^{2}+1\right)}^{2}-x^{2}\right\} \left[সূত্র\ {\left(a+b\right)}^{2}=a^{2}+2ab+b^{2}\right]\\
&=x\left(x-1\right)\left(x^{2}+1+x\right)\left(x^{2}+1-x\right) \left[সূত্র\ {\ \ a}^{2}-b^{2}=\left(a+b\right)\left(a-b\right)\right]\\
&=x\left(x-1\right)\left(x^{2}+x+1\right)\left(x^{2}-x+1\right)\\
&\mathbf{১৫. 4x^{3}-5x^{2}+5x-1}\\
&সমাধানঃ\\
&মনে \text{ } করি, f\left(x\right)=4x^{3}-5x^{2}+5x-1\\
&f\left(\frac{1}{4}\right)=4{\left(\frac{1}{4}\right)}^{3}-5{\left(\frac{1}{4}\right)}^{2}+5\left(\frac{1}{4}\right)-1\\
&= 4.\frac{1}{64}-5\frac{1}{16}+\frac{5}{4}-1\\
&=\frac{1}{16}-\frac{5}{16}+\frac{5}{4}-1\\
&= \frac{1-5+20-16}{16}\\
&= \frac{0}{16}\\
&=0\\
&অতএব, (4x-1) হলো\text{ } f(x)-এর \text{ } একটি \text{ } উৎপাদক।\\
&4x^{3}-5x^{2}+5x-1\\
&= 4x^{3}-x^{2}-4x^{2}+x+4x-1\\
&= x^{2}\left(4x-1\right)-x\left(4x-1\right)+1\left(4x-1\right)\\
&=\left(4x-1\right)\left(x^{2}-x+1\right) [কমন]\\
&\mathbf{১৬. 18x^{3}+15x^{2}-x-2}\\
&সমাধানঃ\\
&মনে \text{ } করি, f\left(x\right)=18x^{3}+15x^{2}-x-2\\
&f\left(-\frac{1}{2}\right)=18{\left(-\frac{1}{2}\right)}^{3}+15{\left(-\frac{1}{2}\right)}^{2}-\left(-\frac{1}{2}\right)-2\\
&= 18.\left(-\frac{1}{8}\right)+15.\frac{1}{4}+\frac{1}{2}-2\\
&=-\frac{9}{4}+\frac{15}{4}+\frac{1}{2}-2\\
&= \frac{-9+15+2-8}{4}\\
&= \frac{0}{4}\\
&=0\\
&অতএব, (2x+1) হলো \text{ } f(x)-এর\text{ } একটি\text{ } উৎপাদক।\\
&18x^{3}+15x^{2}-x-2\\
&=18x^{3}+9x^{2}+6x^{2}+3x-4x-2\\
&=9x^{2}\left(2x+1\right)+3x\left(2x+1\right)-2\left(2x+1\right)\\
&=\left(2x+1\right)\left(9x^{2}+3x-2\right) [কমন]\\
&= \left(2x+1\right)\left(9x^{2}+6x-3x-2\right) [ মিডল টার্ম ব্রেক ]\\
&= \left(2x+1\right)\{3x\left(3x+2\right)-1\left(3x+2\right)\} [কমন]\\
&=\left(2x+1\right)\left(3x+2\right)\left(3x-1\right) [কমন]
\end{align*}