1.Carry out the following divisions.
3.Divide the given polynomial by the given monomial.
(i) \((5x^2 - 6x) ÷ 3x\)
Solution:
(ii) \((3y^8 - 4y^6 + 5y^4) ÷ y^4\)
Solution:
(iii) \(8(x^3y^2z^2 + x^2y^3z^2 + x^2y^2z^3) ÷ 4x2y^2z^2\)
Solution:
(iv) \((x^3 + 2x^2 + 3x) ÷ 2x\)
Solution:
4.Divide as directed.
(i) \(5(2x +1)(3x + 5) ÷ (2x +1) \)
Solution:
(ii) \(26xy(x + 5)(y - 4) ÷ 13x( y - 4)\)
Solution:
(iii) \(52 pqr(p + q)(q + r)(r + p) ÷ 104pq(q + r)(r + p)\)
Solution:
(iv) \(20(y + 4) (y^2 + 5y + 3) ÷ 5(y + 4)\)
Solution:
Solution:
(x + 2)(x + 3)
5.Factorize the expressions and divide them as directed.
(i) \((y^2 + 7y + 10) ÷ (y + 5) \)
Solution:
(i) \[(y^2 + 7y + 10) ÷ (y + 5)\\ (y^2 + 7y + 10) = y^2 + 2 y + 5 y +10 \\= y(y + 2) + 5(y + 2) \\= (y + 2)(y + 5)\]
Thus, \[(y^2 + 7y + 10) ÷ (y + 5) \\= {(y + 2)(y + 5) \over (y + 5) = y + 2} \]
(ii) \((m^2 -14m - 32) ÷ (m + 2) \)
Solutions:
\[(m^2 -14m - 32) ÷ (m + 2)\\ (m^2 -14m - 32)\\ = m^2 + 2m -16m - 32 \\= m(m + 2) -16(m + 2)\\= (m + 2)(m - 16)\]
Thus, \[(m^2 -14m - 32) ÷ (m + 2) = {(m + 2)(m - 16) \over (m + 2)}\\= m - 16 \]
(iii)\( (5p^2 - 25p + 20) ÷ (p -1) \)
Solutions:
\[(5p^2 - 25p + 20) ÷ (p -1)\\ (5p^2 - 25p + 20) = \\5(p^2 - 5p + 4) \\= 5(p^2 - p - 4p + 4) \\= 5[p(p - 1) - 4(p - 1)] \\= 5(p - 1)(p - 4)\]
Thus, \[(5p^2 - 25p + 20) ÷ (p - 1) \\= {5(p - 1)(p - 4) \over (p -1) = 5(p - 4)} \]
(iv) \(4yz(z^2 + 6z -16) ÷ 2y(z + 8) \)
Solutions:
\(4yz(z^2 + 6z -16) ÷ 2y(z + 8) \)
\( 4yz(z^2 + 6z -16) = 4 yz (z^2 - 2z + 8z -16) \)
\(= 4 yz [z(z - 2) + 8(z - 2)] \)
\(= 4 yz(z - 2)(z + 8) \)
Thus, \[4yz(z^2 + 6z -16) ÷ 2y(z + 8)\\= {4 yz(z - 2)(z + 8) \over 2y(z + 8)}\\ = 2z(z - 2) \]
(v) \(5pq(p^2 - q^2) ÷ 2p(p + q) \)
Solutions:
\(5pq(p^2 - q^2) ÷ 2p(p + q)\)
\( 5pq(p^2 - q^2) = 5pq(p - q)(p + q) [ a^2 - b2 = (a + b)(a - b)] \)
Thus, \[5pq(p^2 - q^2) ÷ 2p(p + q) \\= {5pq(p - q)(p + q) \over 2p(p + q)}\\ = {5q(p - q) \over 2} \]
(vi) \(12xy(9x^2 -16y^2) ÷ 4xy(3x + 4y) \)
Solutions:
\(12xy(9x^2 -16y^2) ÷ 4xy(3x + 4y) \)
\( 12xy(9x^2 -16y^2) = 12xy[(3x)^2 - (4y)^2] \)
\(= 12xy(3x - 4y)(3x + 4y) [ a^2 - b^2 = (a + b)(a - b)] \)
\(= 2 × 2 × 3 × x × y × (3x - 4y) × (3x + 4y)\)
Thus, \(12xy(9x^2 -16y^2) ÷ 4xy(3x + 4y)\)
\(= {2 × 2× 3 × x × y × (3x - 4y) × (3x + 4y) \over 4xy(3x + 4y)}\\ = 3(3x - 4y) \)
(vii) \(39y^3(50y^2 - 98) ÷ 26y^2(5y + 7) \)
Solutions:
\(39y^3(50y^2 - 98) ÷ 26y^2(5y + 7) \)
\(39y^3(50y^2 - 98) = 3 × 13 × y × y × y × [2 × (25y^2 - 49)] \)
\(= 3 × 13 × 2 × y × y × y × [(5y)^2 - (7)^2] \)
\[= 3 × 13 × 2 × y × y × y(5y - 7)(5y + 7) [ a^2 - b^2 = (a + b)(a - b)]\]
\(26y^2(5y + 7) \\= 2 × 13 × y × y × (5y + 7) \)
Thus, \(39y^3(50y^2 - 98) ÷ 26y^2(5y + 7) \)
\[= 3 × 13 × 2 × y × y × y(5y - 7)(5y + 7) \over 2 × 13 × y × y × (5y + 7) \\ = 3y(5y - 7)\]
(i) \(28x^4 ÷ 56x\)
Solution:
\(28x^4 ÷ 56x \)
\( 28x^4 = 2 × 2 × 7 × x × x × x × x \)
\(56x =2 × 2 × 2 × 7 × x \)
\[ 28x^4 ÷ 56x = {(2 × 2 × 7 × x × x × x × x) \over (2 × 2 × 2 × 7 × x)}= x^3\over 2 \]
\( 28x^4 = 2 × 2 × 7 × x × x × x × x \)
\(56x =2 × 2 × 2 × 7 × x \)
\[ 28x^4 ÷ 56x = {(2 × 2 × 7 × x × x × x × x) \over (2 × 2 × 2 × 7 × x)}= x^3\over 2 \]
(ii) \(–36y^3 ÷ 9y^2\)
Solution:
\(- 36y^3 ÷ 9y^2 \)
\( -36y^3 = - 2 × 2 × 3 × 3 × y × y × y \)
\( 9y^2= 3 × 3 × y × y \)
\[-36y^3 ÷ 9y^2 = {(-2 × 2 × 3 × 3 × y × y × y)\over (3 × 3 × y × y)}= -4y \]
\( -36y^3 = - 2 × 2 × 3 × 3 × y × y × y \)
\( 9y^2= 3 × 3 × y × y \)
\[-36y^3 ÷ 9y^2 = {(-2 × 2 × 3 × 3 × y × y × y)\over (3 × 3 × y × y)}= -4y \]
(iii) \(66pq^2r^3 ÷ 11qr^2\)
Solution:
\(66 pq^2r^3 ÷ 11qr^2 \)
\( 66pq^2r^3 =2 × 3 × 11 × p × q × q × r × r × r \)
\(11qr^2=11 × q × r × r \)
\[66 pq^2r^3 ÷ 11qr^2 = {(2 × 3 × 11 × p × q × q × r × r × r) \over (11 × q × r × r)}= 6pqr\]
\( 66pq^2r^3 =2 × 3 × 11 × p × q × q × r × r × r \)
\(11qr^2=11 × q × r × r \)
\[66 pq^2r^3 ÷ 11qr^2 = {(2 × 3 × 11 × p × q × q × r × r × r) \over (11 × q × r × r)}= 6pqr\]
(iv) \(34x^3y^3z^3 ÷ 51xy^2z^3\)
Solution:
\(34x^3y^3z^3 ÷ 51xy^2z^3 \)
\(34x^3y^3z^3 =2 × 17 × x × x × x × y × y × y × z × z × z \)
\(51xy^2z^3=3 × 17 × x × y × y × z × z × z \)
\[34x^3y^3z^3 ÷ 51xy^2z^3 = {(2 × 17 × x × x × x × y × y × y × z × z × z) \over (3 × 17 × x × y × y × z × z × z)}= 2x^2y \over 3 \]
\(34x^3y^3z^3 =2 × 17 × x × x × x × y × y × y × z × z × z \)
\(51xy^2z^3=3 × 17 × x × y × y × z × z × z \)
\[34x^3y^3z^3 ÷ 51xy^2z^3 = {(2 × 17 × x × x × x × y × y × y × z × z × z) \over (3 × 17 × x × y × y × z × z × z)}= 2x^2y \over 3 \]
(v) \(12a^8b^8 ÷ (– 6a^6b^4)\)
Solution:
Solution:
12a^8b^8 ÷ (-6a^6b^4)
12a^8b^8 = 2 × 2 × 3 × a^8 × b^8
-6a^6b^4 = -2 × 3 × a^6 × b^4
\[12a^8b^8 ÷ (-6a^6b^4) = {(2 × 2 × 3 × a8 × b8) \over (-2 × 3 × a6 × b4)}= -2a^2b^4\]
12a^8b^8 = 2 × 2 × 3 × a^8 × b^8
-6a^6b^4 = -2 × 3 × a^6 × b^4
\[12a^8b^8 ÷ (-6a^6b^4) = {(2 × 2 × 3 × a8 × b8) \over (-2 × 3 × a6 × b4)}= -2a^2b^4\]
3.Divide the given polynomial by the given monomial.
(i) \((5x^2 - 6x) ÷ 3x\)
Solution:
\((5x^2 - 6x) ÷ 3x\)
\((5x^2 - 6x) = x(5x - 6)\)
Then, \[(5x^2 - 6x) ÷ 3x = {x(5x - 6) \over 3x}\]
\(= {(5x - 6) \over 3}\)
\((5x^2 - 6x) = x(5x - 6)\)
Then, \[(5x^2 - 6x) ÷ 3x = {x(5x - 6) \over 3x}\]
\(= {(5x - 6) \over 3}\)
(ii) \((3y^8 - 4y^6 + 5y^4) ÷ y^4\)
Solution:
\((3y^8 - 4y^6 + 5y^4) ÷ y^4\)
\((3y^8 - 4y^6 + 5y^4) = y^4(3y^4 - 4y^2 + 5)\)
Then, \[(3y^8 - 4y^6 + 5y^4) ÷ y^4 = {y^4(3y^4 - 4y^2 + 5) \over y^4}\]
\(= 3y^4 - 4y^2 + 5\)
\((3y^8 - 4y^6 + 5y^4) = y^4(3y^4 - 4y^2 + 5)\)
Then, \[(3y^8 - 4y^6 + 5y^4) ÷ y^4 = {y^4(3y^4 - 4y^2 + 5) \over y^4}\]
\(= 3y^4 - 4y^2 + 5\)
(iii) \(8(x^3y^2z^2 + x^2y^3z^2 + x^2y^2z^3) ÷ 4x2y^2z^2\)
Solution:
\(8(x^3y^2z^2 + x^2y^3z^2 + x^2y^2z^3) ÷ 4x^2y^2z^2\)
\(8(x^3y^2z^2 + x^2y^3z^2 + x^2y^2z^3) = 8x^2y^2z^2(x + y + z)\)
Then, \[8(x^3y^2z^2 + x^2y^3z^2 + x^2y^2z^3) ÷ 4x^2y^2z^2= {8x^2y^2z^2(x + y + z) \over 4x^2y^2z^2}\]
\(= 2(x + y + z)\)
\(8(x^3y^2z^2 + x^2y^3z^2 + x^2y^2z^3) = 8x^2y^2z^2(x + y + z)\)
Then, \[8(x^3y^2z^2 + x^2y^3z^2 + x^2y^2z^3) ÷ 4x^2y^2z^2= {8x^2y^2z^2(x + y + z) \over 4x^2y^2z^2}\]
\(= 2(x + y + z)\)
(iv) \((x^3 + 2x^2 + 3x) ÷ 2x\)
Solution:
\((x^3 + 2x^2 + 3x) ÷ 2x\)
\((x^3 + 2x^2 + 3x) = x(x^2 + 2x + 3)\)
Then, \[(x^3 + 2x^2 + 3x) ÷ 2x = {x(x^2 + 2x + 3) \over 2x}\]
\(= {(x^2 + 2x + 3) \over 2}\)
\((x^3 + 2x^2 + 3x) = x(x^2 + 2x + 3)\)
Then, \[(x^3 + 2x^2 + 3x) ÷ 2x = {x(x^2 + 2x + 3) \over 2x}\]
\(= {(x^2 + 2x + 3) \over 2}\)
(v) \((p^3q^6 - p^6q^3) ÷ p^3q^3\)
Solution:
Solution:
\((p^3q^6 - p^6q^3) ÷ p^3q^3\)
\((p^3q^6 - p^6q^3) = p^3q^3 (q^3 - p^3)\)
Then, \[(p^3q^6 - p^6q^3) ÷ p^3q^3= {p^3q^3 (q^3 - p^3) \over p^3q^3}\]
\(= q^3 - p^3\)
\((p^3q^6 - p^6q^3) = p^3q^3 (q^3 - p^3)\)
Then, \[(p^3q^6 - p^6q^3) ÷ p^3q^3= {p^3q^3 (q^3 - p^3) \over p^3q^3}\]
\(= q^3 - p^3\)
4.Divide as directed.
(i) \(5(2x +1)(3x + 5) ÷ (2x +1) \)
Solution:
\(5(2x +1)(3x + 5) ÷ (2x +1)\)
\(= {5(2x +1)(3x + 5) \over (2x +1)}\)
\(= 5(3x + 5)\)
\(= {5(2x +1)(3x + 5) \over (2x +1)}\)
\(= 5(3x + 5)\)
Solution:
\(26xy(x + 5)(y - 4) ÷ 13x(y - 4)\)
\(= 2 × 13 × xy(x + 5)(y - 4) \over 13x(y - 4)\)
\(= 2y(x + 5)\)
\(= 2 × 13 × xy(x + 5)(y - 4) \over 13x(y - 4)\)
\(= 2y(x + 5)\)
(iii) \(52 pqr(p + q)(q + r)(r + p) ÷ 104pq(q + r)(r + p)\)
Solution:
\(52pqr (p + q)(q + r)(r + p) ÷ 104pq(q + r)(r + p)\)
\(= 2 × 2 × 13 × p × q × r × ( p + q) × (q + r) × (r + p) \over 2 × 2 × 2 × 13 × p × q × (q + r) × (r + p)\)
\(= r(p + q) \over 2\)
\(= 2 × 2 × 13 × p × q × r × ( p + q) × (q + r) × (r + p) \over 2 × 2 × 2 × 13 × p × q × (q + r) × (r + p)\)
\(= r(p + q) \over 2\)
(iv) \(20(y + 4) (y^2 + 5y + 3) ÷ 5(y + 4)\)
Solution:
\(20(y + 4) (y^2 + 5y + 3) ÷ 5(y + 4)\)
\(= {2 × 2 × 5 × (y + 4) × (y^2 + 5 y + 3) \over 5 × (y + 4)}\)
\(= 4(y^2 + 5 y + 3)\) (v) \(x(x +1)(x + 2)(x + 3) ÷ x(x +1)\)
\(= (x + 2)(x + 3)\)
\(= {2 × 2 × 5 × (y + 4) × (y^2 + 5 y + 3) \over 5 × (y + 4)}\)
\(= 4(y^2 + 5 y + 3)\) (v) \(x(x +1)(x + 2)(x + 3) ÷ x(x +1)\)
\(= (x + 2)(x + 3)\)
(v) \(x(x +1)(x + 2)(x + 3) ÷ x(x +1)\)
Solution:
\(x(x +1)(x + 2)(x + 3) ÷ x(x +1)\)
5.Factorize the expressions and divide them as directed.
(i) \((y^2 + 7y + 10) ÷ (y + 5) \)
Solution:
(i) \[(y^2 + 7y + 10) ÷ (y + 5)\\ (y^2 + 7y + 10) = y^2 + 2 y + 5 y +10 \\= y(y + 2) + 5(y + 2) \\= (y + 2)(y + 5)\]
Thus, \[(y^2 + 7y + 10) ÷ (y + 5) \\= {(y + 2)(y + 5) \over (y + 5) = y + 2} \]
(ii) \((m^2 -14m - 32) ÷ (m + 2) \)
Solutions:
\[(m^2 -14m - 32) ÷ (m + 2)\\ (m^2 -14m - 32)\\ = m^2 + 2m -16m - 32 \\= m(m + 2) -16(m + 2)\\= (m + 2)(m - 16)\]
Thus, \[(m^2 -14m - 32) ÷ (m + 2) = {(m + 2)(m - 16) \over (m + 2)}\\= m - 16 \]
(iii)\( (5p^2 - 25p + 20) ÷ (p -1) \)
Solutions:
\[(5p^2 - 25p + 20) ÷ (p -1)\\ (5p^2 - 25p + 20) = \\5(p^2 - 5p + 4) \\= 5(p^2 - p - 4p + 4) \\= 5[p(p - 1) - 4(p - 1)] \\= 5(p - 1)(p - 4)\]
Thus, \[(5p^2 - 25p + 20) ÷ (p - 1) \\= {5(p - 1)(p - 4) \over (p -1) = 5(p - 4)} \]
(iv) \(4yz(z^2 + 6z -16) ÷ 2y(z + 8) \)
Solutions:
\(4yz(z^2 + 6z -16) ÷ 2y(z + 8) \)
\( 4yz(z^2 + 6z -16) = 4 yz (z^2 - 2z + 8z -16) \)
\(= 4 yz [z(z - 2) + 8(z - 2)] \)
\(= 4 yz(z - 2)(z + 8) \)
Thus, \[4yz(z^2 + 6z -16) ÷ 2y(z + 8)\\= {4 yz(z - 2)(z + 8) \over 2y(z + 8)}\\ = 2z(z - 2) \]
(v) \(5pq(p^2 - q^2) ÷ 2p(p + q) \)
Solutions:
\(5pq(p^2 - q^2) ÷ 2p(p + q)\)
\( 5pq(p^2 - q^2) = 5pq(p - q)(p + q) [ a^2 - b2 = (a + b)(a - b)] \)
Thus, \[5pq(p^2 - q^2) ÷ 2p(p + q) \\= {5pq(p - q)(p + q) \over 2p(p + q)}\\ = {5q(p - q) \over 2} \]
(vi) \(12xy(9x^2 -16y^2) ÷ 4xy(3x + 4y) \)
Solutions:
\(12xy(9x^2 -16y^2) ÷ 4xy(3x + 4y) \)
\( 12xy(9x^2 -16y^2) = 12xy[(3x)^2 - (4y)^2] \)
\(= 12xy(3x - 4y)(3x + 4y) [ a^2 - b^2 = (a + b)(a - b)] \)
\(= 2 × 2 × 3 × x × y × (3x - 4y) × (3x + 4y)\)
Thus, \(12xy(9x^2 -16y^2) ÷ 4xy(3x + 4y)\)
\(= {2 × 2× 3 × x × y × (3x - 4y) × (3x + 4y) \over 4xy(3x + 4y)}\\ = 3(3x - 4y) \)
(vii) \(39y^3(50y^2 - 98) ÷ 26y^2(5y + 7) \)
Solutions:
\(39y^3(50y^2 - 98) ÷ 26y^2(5y + 7) \)
\(39y^3(50y^2 - 98) = 3 × 13 × y × y × y × [2 × (25y^2 - 49)] \)
\(= 3 × 13 × 2 × y × y × y × [(5y)^2 - (7)^2] \)
\[= 3 × 13 × 2 × y × y × y(5y - 7)(5y + 7) [ a^2 - b^2 = (a + b)(a - b)]\]
\(26y^2(5y + 7) \\= 2 × 13 × y × y × (5y + 7) \)
Thus, \(39y^3(50y^2 - 98) ÷ 26y^2(5y + 7) \)
\[= 3 × 13 × 2 × y × y × y(5y - 7)(5y + 7) \over 2 × 13 × y × y × (5y + 7) \\ = 3y(5y - 7)\]
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