সপ্তম শ্রেণীর গণিত উৎপাদকে বিশ্লেষণ ১৯.২ সমাধান /class 7 math 19.2 solution
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সপ্তম শ্রেণীর গণিত সমাধান
উৎপাদকে বিশ্লেষণ
কষে দেখি – 19.2
1. (i) x² + 14x + 49
= (x)² + 2.x.7 + (7)²
= (x + 7)² = (x + 7) (x + 7)
(ii) 4m²-36m +81
(2m)²2.2m.9+ (9)²
= (2m - 9)²
= (2m + 9) (2m - 9)
(iii) 25x² + 30x + 9
(5x)2 + 2.5x.3 + (3)²
= (5x + 3)²
= (5x + 3) (5x + 3)
(iv) 121b²-88bc + 16c²
= (11b)² - 2.11b.4c + (4c)²
= (11b-4c)²
(11b + 4c) (11b-4c)
(v) (x²y)² - 4x²y²
= x² - 4x²y2
x²y²(x² - 4)
= x²y²{(x)² - (2)²}
= x²y² (x + 2) (x - 2)
(vi) a⁴ + 4a²b² + 4b⁴
=(a²)² + 2.a².2b² + (2b²)²
=(a² + 2b²)²
=(a² + 2b²) (a² + 2b²)
(vii) 4x² - 16
=(2x)² – (4)²
= (2x + 4) (2x - 4)
(viii) 121 - 36x²
=(11)² - (6x)²
=(11 + 6x) (11- 6x)
(ix)
(xy)²-(pq)²
(xy + pq) (xy-pq)
(x) 80m² - 125
= 5(16m² - 25)
= 5{(4m)² – (5)²}
=5(4m + 5) (4m - 5)
(xi) ax² - ay²
=a(x² - y²)
=a{(x)² = (y)²}
=a(x + y) (x - y)
(xii) 1 - (m + n)²
= (1)²-(m + n)²
= (1 + m + n) (1 - m - n)
(xiii) (2a - b - c)² - (a - 2b - c)²
= A² - B² [2a-b-c = A, a-2b-c = B]ধরিয়া
=(A + B) (A - B)
={(2a-b-c) + (a-2b-c)} {(2a-b-c) - (a-2b - c)}
={2a-b-c+a-2b-c} {2a - b - ¢ -a + 2b + ¢}
=(3a-3b-2c) (a + b).
(xiv) x² - 2xy - 3y²
= (x)² - 2.x.y + (y)² - 4y²
= (x - y)² - (2y)²
= (x - y + 2y) (x - y - 2y)
= (x +y)(x-3y)
(xvi) a²b²+ 2bc - c²
= (a)²-(b²-2bc + c²)
= (a)²- (b − c)²
= (a + bc) (a - b + c)
(xv) x² + 9y² + 6xy - z²
= (x)² + 2.x.3y + (3y)²-(z)²
= (x + 3y)²-(z)²
= (x + 3y + z) (x + 3y - z)
(xvii) a²(b - c)² - b²(c - a)²
={a(b-c)}2 - {b(c - a)}²
{a(b- c) + b(ca)} {a(b - c) - b(c - a)}
= {ab - ac + bc- ab} {abac - bc + ab}
= (bc - ac) (2ab - ac- bc)
c(ba) (2ab-ac - bc).
(xviii) x² - y²-6yz - 9z²
=x²-(y² + 6yz + 9z²)
= (x)² - {(y)² +2.y.3z + (3z)²)
= (x)²-(y + 3z)² = (x + y + 3z) (x - y - 3z)
(xix) x² - y² + 4x - 4y
= (x + y) (x - y) + 4(x-y
= (x - y) (x + y + 4)
(xx) a²-b²+ c²-d²-2(ac - bd)
=a²-b²+ c²-d² - 2ac + 2bd
= (a²- 2ac + c²) - b² + 2bd - d²
= (a²- 2ac + c²) - (b² - 2bd + d²)
{(a)²-2.a.c +(c)²} - {(b)² - 2.b.d + (d)²}
(a-c)²-(b- d)²
(xxi) 2ab - a² - b² + c²
=c²a²b² + 2ab
=c²(a²- 2ab + b²)
=(c)² - {(a)²-2.a.b + (b)²}
=(c)²(a - b)²
= (c + ab) (c- a + b)
(xxii) 36x² - 16a² - 24ab-9b²
= 36x² - (16a²2 + 24ab + 9b²)
= {(a-c) + (b-d)} {(a -c) - (b-d)}
= (a-c+b-d) (a-c-b+d)
= (a + b-c-d) (a-b-c + d)
= (6x)² - {(4a)2 + 2.4a.3b + (3b)²}
= (6x)²-(4a + 3b)² = (6x + 4a + 3b) (6x - 4a - 3b)
(xiii) a² - 1+2b-b²
= (a)²-(1-2b + b²)
= (a)² - {(1)²-2.1.b + (b)²}
= (a)²-(1- b)²
= (a +1-b) (a - 1 + b)
= (a - b + 1) a + b - 1).
(xxiv) a² - 2a-b² + 2b
=a²b² - 2a + 2b
= (a)²-(b)²-2(a - b)
= (a + b) (a - b)-2(a - b)
= (a - b) {(a + b)-2} = (a - b) (a + b-2).
(xxv) (a²-b²) (c²-d²) -4abcd
=a²c²-b²c²-a²d² + b²d²-4abcd
=a²c²-b²c²-a²d²+ b²d² - 2abcd - 2abcd
a²c²- 2abcd + b²d²-a²d² - 2abcd - b²c²
={(ac)² - 2.ac.bd + (bd)²} - (a²d² + 2abcd + b²c²)
={(ac)²-2.ac.bd + (bd)²} - {(ad)2 + 2.ad.bc + (bc)²}
=(ac - bd)²-(ad + bc)²
=(ac - bd + ad + bc) (ac - bd - ad- bc)
(xxvi) a²-b² - 4ac + 4bc
= (a + b) (a - b) - 4c (a - b)
= (a - b) {(a + b) - 4c} = (a - b) (a + b-4c)
(xxvii) (a²-b²-c² +d²)² - 4(ad - bc)²
(a²-b²c² + d²)² - {2(ad - bc)}²
, a²-b²-c²+ d² = AR2(ad - bc) = B
= A²-B²
(A + B) (A - B)
={(a²-b²-c² + d²) + 2(ad - bc)} {(a²-b²c² + d²) - 2(ad - bc)}
=(a²-b²-c² + d² + 2ad-2bc) (a²-b²-c² + d² - 2ad + 2bc}
={a² + 2ad + d²b² - 2bc-c²} (a²-2ad + d²b² + 2bc-c²}
=[(a)² + 2.a.d + (d)² - {b² + 2bc + c²}] × [(a)² - 2.a.d + (d)² - {b²-2bc + c²}]
=[(a + d)²-(b + c)²] x [(a-d)²-(b-c)²]
={(a + d) + (b + c)} {(a + d) − (b + c)} × {(a − d) + (b-c)} x {(a-d) - (b-c)
=(a+b+c+d) (a+d-b-c) (a-d+b-c) (a-d-b+c)
- (a+b+c+d) (a-b-c+d) (a+b-c-d) (a-b+c-d)
(xviii) 3x² - y² + z²2 - 2xy - 4xz
= - 4x²-x² - y² + z² - 2xy - 4xz
= - 4x² - 4xz + z²-x² - 2xy - y²
= - (4x² - 4x² + z²) - (x² + 2xy + y²)
((2x)² - 2.2x.z+ (2)²) - ((x)² + 2.x.y + (y)²)
(2x - 2)²-(x + y)²
A²-B² [A = 2x - zB = x+y]
= (A + B) (A - B)
= ((2x - 2) + (x + y)) ((2x − 2) - (x + y)
(2x-z + x + y) (2x -z- x - y)
= (3x + y-z) (x-y-z)
2. (1) 81x⁴+ 4y⁴
= (9x²)² + (2y²)²
=(9x²)² + (2y²)² + 2.9x².2y² - 36x²y²
(9x²+ 2y²)²-(6xy)²
= (9x² + 2y² + 6xy) (9x² + 2y² - 6xy)
- (9x² + 6xy + 2y2) (9x² - 6xy + 2y²)
(ii) p⁴- 13p²q2 + 4q¹
(p²)²-2.p².q² + (2q²)² - 9p²q²
(p²-2q²)²-(3pq)²
(p² - 2q² + 3pq) (p² - 2q² - 3pq)
(p² +3pq - 2q²) (p² - 3pq - 2q²).
(iii) x⁸-16y⁸
= (x⁴)²-(4y⁴)²
= (x + 4y⁴) (x - 4y⁴)
°{(x²)² + (2y²)²} {(x²)² - (2y²)²}
={(x²)² + (2y²)² + 2.x².2y2 - 4x²y²} {(x² + 2y²) (x² - 2y²)}
= {(x² + 2y2)² - (2xy)²} (x² + 2y²) (x²-2y2)
= (x² + 2y² + 2xy) (x² + 2y² - 2xy) (x² + 2y²) (x² - 2y²)
=(x² + 2xy + 2y²) (x² - 2xy + 2y²) (x² + 2y²) (x² - 2y²)
(iv) x² + x²y2 + y²
=(x²)² + 2.x²y² + (y²)² - x²y²
= (x² + y²)²-(xy)²
= (x² + y² + xy) (x² + y² − xy)
= (x² + xy + y²) (x² - xy + y²)
(v) 3x⁴ + 2x2y2-y⁴
= 2x⁴ + 2x²y² + x² - y²
2x²(x² + y²) + (x²)² – (y²)²
= 2x²(x² + y²) + (x² + y²) (x² - y²)
= (x² + y²) (2x² + x² - y²)
= (x² + y²) (3x² - y²).
(vi) x^ + x² + 1.
= (x²)² + 2.x².1 + (1)² – x²
= (x² + 1)²-(x)²
= (x² + 1 + x) (x² + 1 - x)
= (x² + x + 1) (x² − x + 1)
(vii) x⁴ + 6x²y² + 8y⁴
= x⁴ + 6x²y² +9y⁴ - y
= {(x²)² + 2.x².3y² + (3y²)²} - y⁴
(x² + 3y²)² = (y²)² = (x² + 3y² + y²) (x² + 3y² - y²)
(x² + 4y²) (x² + 2y²).
(viii) 3x² - y² + z² - 2xy - 4xz
4x²-x² - y² + z² - 2xy - 4xz
= (4x² - 4xz + z²) - (x² + 2xy + y²)
={(2x)² - 2.2x.z + (z)²} - {(x)² + 2.x.y + (y)²}
=(2x - z)²-(x + y)²
=A²-B² [A=2x-zএবংB=x+yধরিয়া
=(A + B) (A - B)
={(2x - z) + (x + y)} {(2x - z) - (x + y)}
=(2x - z + x + y} {2x - z - x - y}
= (3x + y = z) (x - y - z).
(ix) 3x⁴ - 4x²y² + y⁴
=4x⁴-x⁴-4x²y²+y⁴
=4x⁴ -4x²y³+y⁴-x⁴
={(2x²)² - 2.2x².y² + (y²)²} - (x²)²
=(2x² - y²)²-(x²)²
=A²- B² [ A = 2x² - y² 3 B = x²]
= (A + B) (A - B)
={(2x² - y²) + x²} {(2x² - y²) - x²}
=(2x² - y² + x²) (2x² - y² - x²)
=(3x² - y²) (x² - y²)
=(3x² - y²) {(x)² - (y)²}
= (3x² - y²) (x + y) (x - y)
(xi) x⁸ + x⁴y⁴+ y⁸
=(x⁴)²+2.x⁴.y²+(y⁴)²-x⁴y⁴
=(x⁴+y⁴)-(x²y²)²
=(x⁴+y⁴+x²y²)(x⁴+y⁴-x²y²)
={(x²)² + 2.x².y² + (y²)² - x²y2} (x⁴ + y⁴ - x²y²)
={(x² + y²)²-(xy)²} (x⁴+ y⁴ - x²y²)
=(x²+ y² + xy) (x² + y² - xy) (x⁴ + y⁴- x²y²)
=(x² + xy + y²) (x² - xy + y²) (x4 - x²y² + y⁴).
(x) p²-2p²q2 - 15q4
= p2p²q² + q - 16q4
=(p2)2-2.p².q² + (q²)² - (4q²)²
(p²- q²)²-(4q²)²
=A² − B² [ A = p² − q² ® B = 4q²]
= (A + B) (A - B)
= (p² - q² + 4q²) (p² - q² - 4q²)
=(p² +3q²) (p² - 5q²)