📝 ଅନୁଶୀଳନୀ - 3 (a)

❓Question 1: ନିମ୍ନଲିଖିତ ମନୋମିଆଲ୍‌ଗୁଡ଼ିକୁ ସାନରୁ ବଡ଼ ଘାତାଙ୍କ କ୍ରମରେ ସଜାଇ ଲେଖ ।

$1.4y^3, \sqrt{2}y^2, -51, 7y^8, -8y^4, \frac{11}{13}y^9, \sqrt{3}y$

✅ Answer: ମନୋମିଆଲ୍‌ଗୁଡ଼ିକର ଘାତାଙ୍କ ଅନୁସାରେ ସାନରୁ ବଡ଼ କ୍ରମ ହେଉଛି:

$-51, \sqrt{3}y, \sqrt{2}y^2, 1.4y^3, -8y^4, 7y^8, \frac{11}{13}y^9$

❓ Question 2: ନିମ୍ନରେ ପ୍ରଦତ୍ତ ମନୋମିଆଲ୍‌ଗୁଡ଼ିକ ମଧ୍ୟରୁ ସଦୃଶ ମନୋମିଆଲ୍‌ଗୁଡ଼ିକୁ ବାଛି ପୃଥକ ଭାବେ ଲେଖ ।

$12x^2, -3x, \frac{1}{\sqrt{2}}x^3, -5x^2, \frac{x}{7}, 15, \sqrt{3}x^3, 10x^4, \frac{8}{11}$

✅ Answer 2: ସଦୃଶ ମନୋମିଆଲ୍ ଗୁଡ଼ିକ ହେଲା:

• ଧ୍ରୁବକ ପଦ: $15$ ଏବଂ $\frac{8}{11}$

• $x$ ର ପଦ: $-3x$ ଏବଂ $\frac{x}{7}$

• $x^2$ ର ପଦ: $12x^2$ ଏବଂ $-5x^2$

• $x^3$ ର ପଦ: $\frac{1}{\sqrt{2}}x^3$ ଏବଂ $\sqrt{3}x^3$

  (ସୂଚନା: $10x^4$ ର କୌଣସି ସଦୃଶ ପଦ ନାହିଁ)

❓ Question 3: ନିମ୍ନସ୍ଥ ପ୍ରତ୍ୟେକ ପ୍ରକାର ପଲିନୋମିଆଲ୍‌ରୁ ଦୁଇଟି ଲେଖାଏଁ ଉଦାହରଣ ଦିଅ ।

(i) ଶୂନ୍‌ଘାତୀ ପଲିନୋମିଆଲ୍

(ii) ଏକ ପଦବିଶିଷ୍ଟ ଦ୍ୱିଘାତୀ ପଲିନୋମିଆଲ୍

(iii) ଦୁଇ ପଦ ବିଶିଷ୍ଟ ତ୍ରିଘାତୀ ପଲିନୋମିଆଲ୍

(iv) ତିନି ପଦ ବିଶିଷ୍ଟ ଦ୍ୱିଘାତୀ ପଲିନୋମିଆଲ୍

✅ ଉତ୍ତର 3: ଏଠାରେ ପ୍ରତ୍ୟେକର ଦୁଇଟି ସମ୍ଭାବ୍ୟ ଉଦାହରଣ ଦିଆଗଲା:

• (i) ଶୂନ୍‌ଘାତୀ ପଲିନୋମିଆଲ୍ ର ଉଦାହରଣ: $5$ ଏବଂ $-7$

• (ii) ଏକ ପଦବିଶିଷ୍ଟ ଦ୍ୱିଘାତୀ ପଲିନୋମିଆଲ୍ ର ଉଦାହରଣ: $3x^2$ ଏବଂ $-y^2$

• (iii) ଦୁଇ ପଦ ବିଶିଷ୍ଟ ତ୍ରିଘାତୀ ପଲିନୋମିଆଲ୍ ର ଉଦାହରଣ: $x^3 + 2$ ଏବଂ $4y^3 - y$

• (iv) ତିନି ପଦ ବିଶିଷ୍ଟ ଦ୍ୱିଘାତୀ ପଲିନୋମିଆଲ୍ ର ଉଦାହରଣ: $x^2 + 2x + 1$ ଏବଂ $3y^2 - y + 5$

4. ଯୋଗ କର (Addition)

(i) ❓ Question $2y^3 - 3y - 4, 2 - y^3 + 5y$

✅ ଉତ୍ତର (Answer):

$(2y^3 - y^3) + (-3y + 5y) + (-4 + 2)$

$= y^3 + 2y - 2$
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(ii)

❓ ପ୍ରଶ୍ନ (Question): $3x^4 - 2x^3 - 5 + x - 5x^2, 3x^3 + 2x^2 - x^4 - x + 1$

✅ ଉତ୍ତର (Answer):

$(3x^4 - x^4) + (-2x^3 + 3x^3) + (-5x^2 + 2x^2) + (x - x) + (-5 + 1)$

$= 2x^4 + x^3 - 3x^2 - 4$

(iii)

❓ ପ୍ରଶ୍ନ (Question): $\frac{3}{4}x^2 - \frac{4}{5}x - 3, \frac{1}{4}x^2 + \frac{4}{5}x + 2$

✅ ଉତ୍ତର (Answer):

$(\frac{3}{4} + \frac{1}{4})x^2 + (-\frac{4}{5} + \frac{4}{5})x + (-3 + 2)$

$= x^2 - 1$

(iv)

❓ ପ୍ରଶ୍ନ (Question): $2.1x^3 + 3.2x^2 + 5 - 3x, 1.9x^3 - 1.2x^2 + 2x - 1$

✅ ଉତ୍ତର (Answer):

$(2.1 + 1.9)x^3 + (3.2 - 1.2)x^2 + (-3 + 2)x + (5 - 1)$

$= 4x^3 + 2x^2 - x + 4$

(v)

❓ ପ୍ରଶ୍ନ (Question): $\frac{1}{2}z^3 - \frac{3}{2}z^2 + 6z, \frac{1}{2}z^2 - \frac{1}{2}z^3 - 3z - 1, z^3 + 2z^2 + 3z - 4$

✅ ଉତ୍ତର (Answer):

$(\frac{1}{2} - \frac{1}{2} + 1)z^3 + (-\frac{3}{2} + \frac{1}{2} + 2)z^2 + (6 - 3 + 3)z + (-1 - 4)$

$= z^3 + z^2 + 6z - 5$

(vi)

❓ ପ୍ରଶ୍ନ (Question): $8x - 3xy + 2xyz, 2xy - 5x + 3xyz, xy - 3x + 4xyz$

✅ ଉତ୍ତର (Answer):

$(8x - 5x - 3x) + (-3xy + 2xy + xy) + (2xyz + 3xyz + 4xyz)$

$= 9xyz$

(vii)

❓ ପ୍ରଶ୍ନ (Question): $5x^2 - 2xy + y^2, 4xy - 2y^2 - 3x^2, 4y^2 - xy - x^2$

✅ ଉତ୍ତର (Answer):

$(5x^2 - 3x^2 - x^2) + (-2xy + 4xy - xy) + (y^2 - 2y^2 + 4y^2)$

$= x^2 + xy + 3y^2$
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➖ 5. ବିୟୋଗ କର (Subtraction)

(i)

❓ ପ୍ରଶ୍ନ (Question): $6x^3 - 13x^2 + 14$ ରୁ $-x^3 + 2x - 7x^2 + 11$

✅ ଉତ୍ତର (Answer):

$(6x^3 - 13x^2 + 14) - (-x^3 - 7x^2 + 2x + 11)$

$= 7x^3 - 6x^2 - 2x + 3$

(ii)

❓ ପ୍ରଶ୍ନ (Question): $t^4 - 11 + 2t^2 - t^3$ ରୁ $2t^3 - 8t^2 - 10$

✅ ଉତ୍ତର (Answer):

$(t^4 - t^3 + 2t^2 - 11) - (2t^3 - 8t^2 - 10)$

$= t^4 - 3t^3 + 10t^2 - 1$

(iii)

❓ ପ୍ରଶ୍ନ (Question): $\frac{12}{13}y^2 - \frac{5}{13}y^3 - 15$ ରୁ $-\frac{1}{13}y^2 + \frac{8}{13}y^3 + 20$

✅ ଉତ୍ତର (Answer):

$(-\frac{5}{13}y^3 + \frac{12}{13}y^2 - 15) - (\frac{8}{13}y^3 - \frac{1}{13}y^2 + 20)$

$= -y^3 + y^2 - 35$

(iv)

❓ ପ୍ରଶ୍ନ (Question): $2.5x^3 - 7 - 3.5x^2$ ରୁ $2.5x^2 + 1.5x^3 + 8 - 2x$
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✅ ଉତ୍ତର (Answer):

$(2.5x^3 - 3.5x^2 - 7) - (1.5x^3 + 2.5x^2 - 2x + 8)$

$= x^3 - 6x^2 + 2x - 15$

(v)

❓ ପ୍ରଶ୍ନ (Question): $x^2 - 2xy + 3y^2$ ରୁ $2x^2 - xy - 2y^2$

✅ ଉତ୍ତର (Answer):

$(x^2 - 2xy + 3y^2) - (2x^2 - xy - 2y^2)$

$= -x^2 - xy + 5y^2$

(vi)

❓ ପ୍ରଶ୍ନ (Question): $2x^2 - 3xy - 4y^2$ ରୁ $x^2 - xy - 2y^2$

✅ ଉତ୍ତର (Answer):

$(2x^2 - 3xy - 4y^2) - (x^2 - xy - 2y^2)$

$= x^2 - 2xy - 2y^2$

(vii)

❓ ପ୍ରଶ୍ନ (Question): $a - 3b + 2c$ ରୁ $3b - 7c + 2a$

✅ ଉତ୍ତର (Answer):

$(a - 3b + 2c) - (2a + 3b - 7c)$

$= -a - 6b + 9c$

(viii)

❓ ପ୍ରଶ୍ନ (Question): $\frac{1}{2}a + \frac{2}{3}b - \frac{3}{2}c$ ରୁ $a - \frac{1}{3}b + \frac{1}{2}c$

✅ ଉତ୍ତର (Answer):

$(\frac{1}{2}a + \frac{2}{3}b - \frac{3}{2}c) - (a - \frac{1}{3}b + \frac{1}{2}c)$

$= -\frac{1}{2}a + b - 2c$

✖️ 6. ନିମ୍ନରେ ଦତ୍ତ ପଲିନୋମିଆଲ୍‌ଗୁଡ଼ିକର ଗୁଣଫଳ ସ୍ଥିର କରି ଗୁଣଫଳର ଘାତ ନିରୂପଣ କର ।

(i)

❓ Question: $2x^2 - 3x + 5$ ଓ $x^2 + 5x + 2$

✅ Answer:

ଗୁଣଫଳ (Product) = $(2x^2 - 3x + 5)(x^2 + 5x + 2)$

$= 2x^4 + 10x^3 + 4x^2 - 3x^3 - 15x^2 - 6x + 5x^2 + 25x + 10$

$= 2x^4 + 7x^3 - 6x^2 + 19x + 10$

ଘାତ (Degree) = $4$

(ii)

❓ Question: $y^3 - 5y^2 + 11y$ ଓ $y^5 - 20y^4 + 17$

✅ Answer:

ଗୁଣଫଳ (Product) = $(y^3 - 5y^2 + 11y)(y^5 - 20y^4 + 17)$

$= y^8 - 20y^7 + 17y^3 - 5y^7 + 100y^6 - 85y^2 + 11y^6 - 220y^5 + 187y$

$= y^8 - 25y^7 + 111y^6 - 220y^5 + 17y^3 - 85y^2 + 187y$

ଘାତ (Degree) = $8$

(iii)
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❓ Question: $(2x+3)$ ଓ $5x^2 - 7x + 8$

✅ Answer:

ଗୁଣଫଳ (Product) = $(2x+3)(5x^2 - 7x + 8)$

$= 10x^3 - 14x^2 + 16x + 15x^2 - 21x + 24$

$= 10x^3 + x^2 - 5x + 24$

ଘାତ (Degree) = $3$

(iv)

❓ Question: $(x-1), (7x-9)$ ଓ $3x^3 - 14x^2 + 8$

✅ Answer:

ଗୁଣଫଳ (Product) = $(x-1)(7x-9)(3x^3 - 14x^2 + 8)$

$= (7x^2 - 16x + 9)(3x^3 - 14x^2 + 8)$

$= 21x^5 - 98x^4 + 56x^2 - 48x^4 + 224x^3 - 128x + 27x^3 - 126x^2 + 72$

$= 21x^5 - 146x^4 + 251x^3 - 70x^2 - 128x + 72$

ଘାତ (Degree) = $5$

(v)

❓ Question: $(x^2 + y^2)$ ଓ $(x^4 - x^2y^2 + y^4)$

✅ Answer:

ଗୁଣଫଳ (Product) = $(x^2 + y^2)(x^4 - x^2y^2 + y^4)$

$= x^6 + y^6$ (ସୂତ୍ର ଅନୁଯାୟୀ)

ଘାତ (Degree) = $6$

(vi)

❓ Question: $(2x+3y), (2x-3y)$ ଓ $(4x^2 + 9y^2)$

✅ Answer:

ଗୁଣଫଳ (Product) = $(2x+3y)(2x-3y)(4x^2 + 9y^2)$

$= (4x^2 - 9y^2)(4x^2 + 9y^2)$

$= 16x^4 - 81y^4$

ଘାତ (Degree) = $4$

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7. ଭାଗଫଳ ଓ ଭାଗଶେଷ ନିରୂପଣ କର ।

(i)

❓ Question: $(x^3 - 1) \div (x - 1)$

✅ Answer:

$$\begin{array}{rll} & x^2 + x + 1 \\ \hline x - 1 \ \vert{} & x^3 - 1 \\ & -(x^3 - x^2) \\ \hline & \quad \ \ x^2 - 1 \\ & \quad -(x^2 - x) \\ \hline & \qquad \quad \ \ x - 1 \\ & \qquad -(x - 1) \\ \hline & \qquad \qquad \ \ 0 \end{array}$$

$\therefore$ ଭାଗଫଳ = $x^2 + x + 1$ and ଭାଗଶେଷ = $0$

(ii)

❓ Question: $(-81y^2 + 64) \div (8 - 9y)$

(Rearranging the dividend: $64 - 81y^2 \div 8 - 9y$)

✅ Answer:

$$\begin{array}{rll} & 8 + 9y \\ \hline 8 - 9y \ \vert{} & 64 - 81y^2 \\ & -(64 - 72y) \\ \hline & \quad \ \ \ \ 72y - 81y^2 \\ & \quad -(72y - 81y^2) \\ \hline & \qquad \qquad \qquad 0 \end{array}$$

$\therefore$ ଭାଗଫଳ= $8 + 9y$ and ଭାଗଶେଷ= $0$

(iii)

❓ Question: $(2x^3 - 7x^2 - x + 2) \div (x^2 - 3x - 2)$

✅ Answer:

$$\begin{array}{rll} & 2x - 1 \\ \hline x^2 - 3x - 2 \ \vert{} & 2x^3 - 7x^2 - x + 2 \\ & -(2x^3 - 6x^2 - 4x) \\ \hline & \quad \ \ -x^2 + 3x + 2 \\ & \quad -(-x^2 + 3x + 2) \\ \hline & \qquad \qquad \qquad \quad 0 \end{array}$$

$\therefore$ ଭାଗଫଳ = $2x - 1$ and ଭାଗଶେଷ = $0$

(iv)

❓ Question: $(x^3 - 14x^2 + 37x - 26) \div (x - 2)$

✅ Answer:

$$\begin{array}{rll} & x^2 - 12x + 13 \\ \hline x - 2 \ \vert{} & x^3 - 14x^2 + 37x - 26 \\ & -(x^3 - 2x^2) \\ \hline & \quad \ \ -12x^2 + 37x \\ & \quad -(-12x^2 + 24x) \\ \hline & \qquad \qquad \ \ 13x - 26 \\ & \qquad \quad \ -(13x - 26) \\ \hline & \qquad \qquad \qquad \quad \ \ 0 \end{array}$$

$\therefore$ ଭାଗଫଳ = $x^2 - 12x + 13$ and ଭାଗଶେଷ = $0$

(v)

❓ Question: $(t^3 - 6t^2 + 11t - 6) \div (t^2 - 5t + 6)$

✅ Answer:

$$\begin{array}{rll} & t - 1 \\ \hline t^2 - 5t + 6 \ \vert{} & t^3 - 6t^2 + 11t - 6 \\ & -(t^3 - 5t^2 + 6t) \\ \hline & \quad \ \ -t^2 + 5t - 6 \\ & \quad -(-t^2 + 5t - 6) \\ \hline & \qquad \qquad \qquad \quad 0 \end{array}$$

$\therefore$ଭାଗଫଳ= $t - 1$ and ଭାଗଶେଷ = $0$

(vi)

❓ Question: $(8a^2 - 34ab + 21b^2) \div (4a + 3b)$

✅ Answer:

$$\begin{array}{rll} & 2a - 10b \\ \hline 4a + 3b \ \vert{} & 8a^2 - 34ab + 21b^2 \\ & -(8a^2 + 6ab) \\ \hline & \quad \ \ -40ab + 21b^2 \\ & \quad -(-40ab - 30b^2) \\ \hline & \qquad \qquad \quad \ \ 51b^2 \end{array}$$

$\therefore$ ଭାଗଫଳ = $2a - 10b$ and ଭାଗଶେଷ = $51b^2$

(vii)

❓ Question: $(16xy^2 - 21x^2y + 9x^3 - 4y^3) \div (x - y)$

(Rearranging the dividend by power: $9x^3 - 21x^2y + 16xy^2 - 4y^3 \div x - y$)

✅ Answer:

$$\begin{array}{rll} & 9x^2 - 12xy + 4y^2 \\ \hline x - y \ \vert{} & 9x^3 - 21x^2y + 16xy^2 - 4y^3 \\ & -(9x^3 - 9x^2y) \\ \hline & \quad \ \ -12x^2y + 16xy^2 \\ & \quad -(-12x^2y + 12xy^2) \\ \hline & \qquad \qquad \quad \ \ 4xy^2 - 4y^3 \\ & \qquad \qquad \ -(4xy^2 - 4y^3) \\ \hline & \qquad \qquad \qquad \qquad \quad \ \ 0 \end{array}$$

$\therefore$ ଭାଗଫଳ = $9x^2 - 12xy + 4y^2$ and ଭାଗଶେଷ = $0$

(viii)

❓ Question: $(x^4 + x^2y^2 + y^4) \div (x^2 - xy + y^2)$

✅ Answer:

$$\begin{array}{rll} & x^2 + xy + y^2 \\ \hline x^2 - xy + y^2 \ \vert{} & x^4 + x^2y^2 + y^4 \\ & -(x^4 - x^3y + x^2y^2) \\ \hline & \quad \ \ \ \ \ x^3y + y^4 \\ & \quad -(x^3y - x^2y^2 + xy^3) \\ \hline & \qquad \qquad \ \ x^2y^2 - xy^3 + y^4 \\ & \qquad \quad -(x^2y^2 - xy^3 + y^4) \\ \hline & \qquad \qquad \qquad \qquad \qquad \ \ 0 \end{array}$$

$\therefore$ ଭାଗଫଳ = $x^2 + xy + y^2$ and ଭାଗଶେଷ = $0$

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📝 8.

❓ Question: ଯଦି $p(x) = 3x^3 - 6x^2 + 2$ ଏବଂ
$q(x) = 2x^2 - 5x + 1$ ତେବେ

(i) $2p(x) - 5q(x)$ ଓ
(ii) $4p(x) + 3q(x)$ ର ମାନ ସ୍ଥିର କର ।

✅ Answer:

ଦତ୍ତ (Given):

$p(x) = 3x^3 - 6x^2 + 2$

$q(x) = 2x^2 - 5x + 1$

(i) $2p(x) - 5q(x)$

$$= 2(3x^3 - 6x^2 + 2) - 5(2x^2 - 5x + 1)$$

$$= 6x^3 - 12x^2 + 4 - 10x^2 + 25x - 5$$

$$= 6x^3 - 22x^2 + 25x - 1$$

(ii) $4p(x) + 3q(x)$

$$= 4(3x^3 - 6x^2 + 2) + 3(2x^2 - 5x + 1)$$

$$= 12x^3 - 24x^2 + 8 + 6x^2 - 15x + 3$$

$$= 12x^3 - 18x^2 - 15x + 11$$

🔍 9.

❓ Question: ଯଦି $p(x) = 2x^3 + 3x + 5$,

$q(x) = x^2 + 4x + 1$ ଓ $r(x) = x - 1$ ହୁଏ ତେବେ ଦର୍ଶାଅ ଯେ,

(i) $p(x) \times q(x) = q(x) \times p(x)$

(ii) $p(x) \times \{q(x) + r(x)\} = p(x) \cdot q(x) + p(x) \cdot r(x)$

✅ Answer:

(i) $p(x) \times q(x) = q(x) \times p(x)$

ବାମପକ୍ଷ (LHS) = $p(x) \times q(x)$

$$= (2x^3 + 3x + 5)(x^2 + 4x + 1)$$

$$= 2x^5 + 8x^4 + 2x^3 + 3x^3 + 12x^2 + 3x + 5x^2 + 20x + 5$$

$$= 2x^5 + 8x^4 + 5x^3 + 17x^2 + 23x + 5$$

ଦକ୍ଷିଣପକ୍ଷ (RHS) = $q(x) \times p(x)$

$$= (x^2 + 4x + 1)(2x^3 + 3x + 5)$$

$$= 2x^5 + 3x^3 + 5x^2 + 8x^4 + 12x^2 + 20x + 2x^3 + 3x + 5$$

$$= 2x^5 + 8x^4 + 5x^3 + 17x^2 + 23x + 5$$

$\therefore$ ବାମପକ୍ଷ (LHS) = ଦକ୍ଷିଣପକ୍ଷ (RHS) (ପ୍ରମାଣିତ / Proved)

(ii) $p(x) \times \{q(x) + r(x)\} = p(x) \cdot q(x) + p(x) \cdot r(x)$

ବାମପକ୍ଷ (LHS) = $p(x) \times \{q(x) + r(x)\}$

$$= (2x^3 + 3x + 5) \times \{(x^2 + 4x + 1) + (x - 1)\}$$

$$= (2x^3 + 3x + 5) \times \{x^2 + 5x\}$$

$$= 2x^5 + 10x^4 + 3x^3 + 15x^2 + 5x^2 + 25x$$

$$= 2x^5 + 10x^4 + 3x^3 + 20x^2 + 25x$$

ଦକ୍ଷିଣପକ୍ଷ (RHS) = $p(x) \cdot q(x) + p(x) \cdot r(x)$

[ପୂର୍ବ (i) ନମ୍ବର ସମାଧାନରୁ $p(x) \cdot q(x) = 2x^5 + 8x^4 + 5x^3 + 17x^2 + 23x + 5$]

ଏବେ $p(x) \cdot r(x)$ ନିର୍ଣ୍ଣୟ କରିବା:

$$= (2x^3 + 3x + 5)(x - 1)$$

$$= 2x^4 - 2x^3 + 3x^2 - 3x + 5x - 5$$

$$= 2x^4 - 2x^3 + 3x^2 + 2x - 5$$

ବର୍ତ୍ତମାନ ଯୋଗ କଲେ (Adding both results):

$$= (2x^5 + 8x^4 + 5x^3 + 17x^2 + 23x + 5) + (2x^4 - 2x^3 + 3x^2 + 2x - 5)$$

$$= 2x^5 + (8+2)x^4 + (5-2)x^3 + (17+3)x^2 + (23+2)x + (5-5)$$

$$= 2x^5 + 10x^4 + 3x^3 + 20x^2 + 25x$$

$\therefore$ ବାମପକ୍ଷ (LHS) = ଦକ୍ଷିଣପକ୍ଷ (RHS) (ପ୍ରମାଣିତ / Proved)

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✏️ 10. ସରଳ କର (Simplify)

(i)

❓ Question: $(x^2 - 3x + 5) + (2x^2 - x - 2) - (3x^2 + 7x - 3)$

✅ Answer:

$$= x^2 - 3x + 5 + 2x^2 - x - 2 - 3x^2 - 7x + 3$$

$$= (x^2 + 2x^2 - 3x^2) + (-3x - x - 7x) + (5 - 2 + 3)$$

$$= 0x^2 - 11x + 6$$

$$= -11x + 6$$

(ii)

❓ Question: $(x^2 - xy + 2y^2) - (2x^2 + 4xy + 3y^2) + (4x^2 - 2xy - y^2)$

✅ Answer:

$$= x^2 - xy + 2y^2 - 2x^2 - 4xy - 3y^2 + 4x^2 - 2xy - y^2$$

$$= (x^2 - 2x^2 + 4x^2) + (-xy - 4xy - 2xy) + (2y^2 - 3y^2 - y^2)$$

$$= 3x^2 - 7xy - 2y^2$$

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(iii)

❓ Question: $(a + b + c)(a - b + c) - (a + b - c)(a - b - c)$

✅ Answer:

ସହଜରେ ଗୁଣନ କରିବା ପାଇଁ ଏହାକୁ ସୂତ୍ର ଅନୁଯାୟୀ ସଜାଇଲେ (Grouping terms):

$$= \{(a + c) + b\}\{(a + c) - b\} - \{a + (b - c)\}\{a - (b - c)\}$$

[ସୂତ୍ର ବ୍ୟବହାର: $(x+y)(x-y) = x^2 - y^2$]

$$= \{(a + c)^2 - b^2\} - \{a^2 - (b - c)^2\}$$

$$= \{a^2 + c^2 + 2ac - b^2\} - \{a^2 - (b^2 + c^2 - 2bc)\}$$

$$= a^2 + c^2 + 2ac - b^2 - a^2 + b^2 + c^2 - 2bc$$

$$= (a^2 - a^2) + (-b^2 + b^2) + (c^2 + c^2) + 2ac - 2bc$$

$$= 2c^2 + 2ac - 2bc$$

(କିମ୍ବା ଏହାକୁ କମନ୍ ନେଇ $2c(c + a - b)$ ମଧ୍ୟ ଲେଖାଯାଇପାରିବ)