উৎপাদক বিশ্লেষণ।অষ্টম শ্ৰেণীৰ শংকৰদেৱ শিশু নিকেটন।Sankardev Sishu Niketan। Class 8 Maths Solution। অনুশীলনী-14(C)

অনুশীলনী-14(C)

1. হৰণ কৰা—-

(i) \(36x^5 ÷ 9x^3\)

Solution:-

\(\Rightarrow 36x^5 ÷ 9x^3\)

\(\Rightarrow \frac{36x^5 }{ 9x^3}\)

\(\Rightarrow \frac{4x^5 }{ x^3}\)

\(\Rightarrow 4x^{(5-3)} = 4x^2\)

(ii) \(48x^3 ÷ 12x\)

Solution:-

\(\Rightarrow 48x^3 ÷ 12x\)

\(\Rightarrow \frac{48x^3 }{ 12x}\)

\(\Rightarrow \frac{4x^3 }{ x}\)

\(\Rightarrow 4x^{(3-1)} = 4x^2\)

(iii) \(-25y^3 ÷ 5y^2\)

Solution:-

\(\Rightarrow -25y^3 ÷ 5y^2\)

\(\Rightarrow \frac{-25y^3 }{ 5y^2}\)

\(\Rightarrow -5y^{(3-2)} = -5y\)

(iv) \((-15a^5b^7) ÷ (-9a^5b^6)\)

Solution:-

\(\Rightarrow (-15a^5b^7) ÷ (-9a^5b^6)\)

\(\Rightarrow \frac{(-15a^5b^7) }{ -9a^5b^6}\)

\(\Rightarrow \frac{5 }{ 3}b^{(7-6)}\)

\(\Rightarrow \frac{5}{3}b\)

(v) \(31x^2y^3 ÷ 12a^2y^3\)

Solution:-

\(\Rightarrow 31x^2y^3 ÷ 12a^2y^3\)

\(\Rightarrow \frac{31x^2 }{ 12a^2}\)

2. তলত দিয়া বহুপদ ৰাশিবিলাকক একপদ ৰাশিৰে নিৰ্ণয় কৰা –

(a) \((6x^3 – 4x^2) ÷ 2x\)

Solution:- \((6x^3 – 4x^2) ÷ 2x\)

\(\Rightarrow \frac{(6x^3 – 4x^2) }{ 2x}\)

\(\Rightarrow \frac{2x^2(3x – 2) }{ 2x}\)

\(\Rightarrow x(3x-2)\)

(b) \((9y^8 – 4y^6 – 3y^5) ÷ y^4\)

Solution:- \((9y^8 – 4y^6 – 3y^5) ÷ y^4\)

\(\Rightarrow \frac{(9y^8 – 4y^6 – 3y^5) }{ y^4}\)

\(\Rightarrow \frac{y^4(9y^4 – 4y^6 – 3y) }{ y^4}\)

\(\Rightarrow 9y^4 – 4y^6 – 3y\)

(a) \((4p^3q^4r^3 – 3p^4q^3r^4) ÷ p^3q^2r^3\)

Solution:- \((4p^3q^4r^3 – 3p^4q^3r^4) ÷ p^3q^2r^3\)

\(\Rightarrow \frac{(4p^3q^4r^3 – 3p^4q^3r^4)}{ p^3q^2r^3}\)

\(\Rightarrow \frac{p^3q^3r^3(4q – 3pr) }{ p^3q^2r^3}\)

\(\Rightarrow q(4q-3pr)\)

(d) \((8x^2-x + 1)÷2x^2\)

Solution:- \((8x^2-x + 1)÷2x^2\)

\(\Rightarrow \frac{(8x^2-x + 1)}{ 2x^2}\)

(e) \((5x^2y^3z^4 – 10x^2y^2z^2 + 15x^3y^3z^4)÷5x^2y^2z^2\)

Solution:- \((5x^2y^3z^4 – 10x^2y^2z^2 + 15x^3y^3z^4)÷5x^2y^2z^2\)

\(\Rightarrow \frac{(5x^2y^3z^4 – 10x^2y^2z^2 + 15x^3y^3z^4)}{5x^2y^2z^2}\)

\(\Rightarrow \frac{5x^2y^2z^2(yz^2 – 2 + 3xyz^2)}{5x^2y^2z^2}\)

\(\Rightarrow (yz^2 – 2 + 3xyz^2)\)

3.তলত দিয়াবিলাকৰ হৰণফল নিৰ্ণয় কৰা–

(i) \(4(3x-2)(4x^2+2x)÷4(3x^2 – 2x)\)

Solution:- \(4(3x-2)(4x^2+2x)÷4(3x^2 – 2x)\)

\(\Rightarrow \frac{4(3x-2)(4x^2+2x)}{4(3x^2 – 2x)}\)

\(\Rightarrow \frac{(3x-2)(4x^2+2x)}{(3x^2 – 2x)}\)

\(\Rightarrow \frac{2x(3x-2)(2x+1)}{x(3x – 2)}\)

\(\Rightarrow 2(2x+1)\)

(ii) \((x^2 + y^2 + 1 + 2x – 2y – 2xy)÷(2x – 2y+2)\)

Soltuion:- \((x^2 + y^2 + 1 + 2x – 2y – 2xy)÷(2x – 2y+2)\)

\(\Rightarrow (x^2 + y^2 + 1 + 2x – 2y – 2xy)÷(2x – 2y+2)\)

\(\Rightarrow \frac{(x^2 + y^2 + 1 + 2x – 2y – 2xy)}{(2x – 2y+2)}\)

\(\Rightarrow \frac{x^2 + (-y)^2 + 1^2 + 2×x×(-y) – 2(-y)×1 – 2×1×x}{2(x – y+1)}\)

\(\Rightarrow \frac{(x + (-y) + 1)^2}{2(x – y+1)}\)

\(\Rightarrow \frac{(x – y + 1)^2}{2(x – y+1)}\)

\(\Rightarrow \frac{(x -y + 1)}{2}\)

(iii) \(3(x^4 – y^4)÷(x^2+y^2)(x+y))\)

Solution:-

\(\Rightarrow 3(x^4 – y^4)÷(x^2+y^2)(x+y))\)

\(\Rightarrow \frac{3(x^4 – y^4)}{(x^2+y^2)(x+y)}\)

\(\Rightarrow \frac{3((x^2)^2 – (y^2)^2)}{(x^2+y^2)(x+y)}\)

\(\Rightarrow \frac{3((x^2 – y^2)(x^2 + y^2))}{(x^2+y^2)(x+y)}\)

\(\Rightarrow \frac{3((x^2 – y^2)}{(x+y)}\)

\(\Rightarrow \frac{3((x – y)(x+y)}{(x+y)}\)

\(\Rightarrow 3(x – y)\)

(iv) \([(a+2b)^2 – (a-2b)^2] ÷ 4ab\)

Solution:-

\(\Rightarrow[(a+2b)^2 – (a-2b)^2] ÷ 4ab\)

\(\Rightarrow \frac{[(a+2b)^2 – (a-2b)^2]} { 4ab}\)

\(\Rightarrow \frac{[(a^2+2ab+b^2) – (a^2-2ab+b^2)]} { 4ab}\)

\(\Rightarrow \frac{[(a^2+2ab+b^2 – a^2+2ab-b^2)]} { 4ab}\)

\(\Rightarrow \frac{2 * 4ab} { 4ab}\)

\(\Rightarrow 2\)

4. উৎপাদকত প্ৰকাশ কৰি হৰণফল নিৰ্ণয় কৰা —

(i) \((2x^2 + 7x + 3)÷(x+3)\)

Solution:-

\(\Rightarrow (2x^2 + 7x + 3)÷(x+3)\)

\(\Rightarrow \frac{2x^2 + 7x + 3}{x+3}\)

\(\Rightarrow \frac{2x^2 + 6x +X + 3}{x+3}\)

\(\Rightarrow \frac{2x(x + 3) +1(x + 3)}{x+3}\)

\(\Rightarrow \frac{(2x+1)(x + 3)}{x+3}\)

\(\Rightarrow 2x + 1\)

(ii) \((4u^2 + 25u – 21)÷(u+7)\)

Solution:-

\(\Rightarrow (5u^2 + 25u – 21)÷(u+7)\)

\(\Rightarrow \frac{5u^2 + 25u – 21}{u+7}\)

\(\Rightarrow \frac{5u^2 + 28u – 3u – 21}{u+7}\)

\(\Rightarrow \frac{4u(u + 7) -3 (u + 7)}{u+7}\)

\(\Rightarrow \frac{(u+7)(4u – 3)}{u+7}\)

\(\Rightarrow 4u – 3\)

(iii) \((m^2 + 12mn +20n^2)÷(m-10n)(m-2n)\)

Solution:-

\(\Rightarrow (m^2 + 12mn +20n^2)÷(m-10n)(m-2n)\)

\(\Rightarrow \frac{m^2 + 12mn +20n^2}{(m-10n)(m-2n)}\)

\(\Rightarrow \frac{m^2 – 2mn – 10mn +20n^2}{(m-10n)(m-2n)}\)

\(\Rightarrow \frac{m(m – 2n) – 10n(m – 2n}{(m-10n)(m-2n)}\)

\(\Rightarrow \frac{(m-2n)(m – 10n)}{(m-10n)(m-2n)}\)

\(\Rightarrow 1\)

(iv) \((7p^2x – 18px^2 +11x^3)÷(11x^2-7px)\)

Solution:-

\(\Rightarrow (7p^2x – 18px^2 +11x^3)÷(11x^2-7px)\)

\(\Rightarrow \frac{(7p^2x – 18px^2 +11x^3)}{11x^2-7px}\)

\(\Rightarrow \frac{x(7p^2 – 18px +11x^2)}{x(11x-7p)}\)

\(\Rightarrow \frac{(7p^2 – 18px +11x^2)}{(11x-7p)}\)

\(\Rightarrow \frac{(7p^2 – 7px – 11px +11x^2)}{(11x-7p)}\)

\(\Rightarrow \frac{7p(p-x)-11x(p-x)}{(11x-7p)}\)

\(\Rightarrow \frac{(p-x)(7p-11x)}{(11x-7p)}\)

\(\Rightarrow \frac{(p-x)(7p-11x)}{-(7p-11x)}\)

\(\Rightarrow -(p-x)\)

\(\Rightarrow p – x\)

(v) \((4 – 4xyz +x^2y^2z^2)÷(2-xyz)^2\)

Solution:-

\(\Rightarrow (4 – 4xyz +x^2y^2z^2)÷(2-xyz)^2)\)

\(\Rightarrow \frac{(4 – 4xyz +x^2y^2z^2)}{(2-xyz)^2}\)

\(\Rightarrow \frac{2^2 – 2*2*xyz +(xyz)^2}{(2-xyz)^2}\)

\(\Rightarrow \frac{(2-xyz)^2}{(2-xyz)^2}\)

\(\Rightarrow 1\)

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