**Factors of Algebraic Expressions – Maharashtra Board Class 7 Solutions for Mathematics (English Medium)**

MathematicsGeneral ScienceMaharashtra Board Solutions

**Exercise 76:**

**Solution 1:**

- 7p = 7 × p

∴ Factors of 7p are 7 and p. - 6m = 6 × m = 2 × 3 × m

∴ Factors of 6m are 2, 3, and m. - 9xy = 9 × x × y = 3 × 3 × x × y

∴ Factors of 9xy are 3, 3, x, and y. - 22ab = 22 × a × b = 2 × 11 × a × b

∴ Factors of 22ab are 2, 11, a, and b. - p
^{2}q = p × p × q

∴ Factors of p^{2}q are p, p, and q. - 10xy
^{2}= 2 × 5 × x × y × y

∴ Factors of 10xy^{2}are 2, 5, x, y, and y. - 5a
^{2}= 5 × a × a

∴ Factors of 5a^{2}are 5, a, and a. - 15m
^{2}n = 3 × 5 × m × m × n

∴ Factors of 15m^{2}n are 3, 5, m, m, and n. - 30a
^{2}b^{2}= 2 × 3 × 5 × a × a × b × b

∴ Factors of 30a^{2}b^{2}are 2, 3, 5, a, a, b, and b. - 12x
^{3}= 2 × 2 × 3 × x × x × x

∴ Factors of 12x^{3}are 2, 2, 3, x, x, and x.

**Exercise 77:**

**Solution 1:**

- 8m, 4m
^{2}n

8m = 2 × 2 × 2 × m

4m^{2}n = 2 × 2 × m × m × n - 3x
^{2}y, 12xy^{2 }3x^{2}y = 3 × x × x × y

12xy^{2}= 2 × 2 × 3 × x × y × y - 15a
^{2}bc, 5ab, 20abc^{2 }15a^{2}bc = 3 × 5 × a × a × b × c

5ab = 5 × a × b

20abc^{2}= 2 × 2 × 5 × a × b × c × c

**Solution 2:**

- 4p
^{2}q, 16pq^{2 }4p^{2}q = 2 × 2 × p × p × q

16pq^{2}= 2 × 2 × 2 × 2 × p × q × q

∴ Common factors = 2 × 2 × p × q = 4pq - 18x
^{3}y^{2}, 12x^{3}y

18x^{3}y^{2}= 2 × 3 × 3 × x × x × x × y × y

12x^{3}y = 2 × 2 × 3 × x × x × x × y

∴ Common factors = 2 × 3 × x × x × x × y = 6x^{3}y - 7a
^{3}b^{2}c, 28a^{2}bc^{2 }7a^{3}b^{2}c = 7 × a × a × a × b × b × c

28a^{2}bc^{2}= 2 × 2 × 7 × a × a × b × c × c

∴ Common factors = 7 × a × a × b × c = 7a^{2}bc - 8x
^{3}y^{2}, 10x^{2}y^{3}, 6x^{2}y

8x^{3}y^{2}= 2 × 2 × 2 × x × x × x × y × y

10x^{2}y^{3}= 2 × 5 × x × x × y × y × y

6x^{2}y = 2 × 3 × x × x × y

∴ Common factors = 2 × x × x × y = 2x^{2}y - 24mnp
^{2}, 22m^{2}p^{2}, 30m^{2}n^{2}p

24mnp^{2}= 2 × 2 × 2 × 3 × m × n × p × p

22m^{2}p^{2}= 2 × 11 × m × m × p × p

30m^{2}n^{2}p = 2 × 3 × 5 × m × m × n × n × p

∴ Common factors = 2 × m × p = 2mp

**Solution 3:**

- 6m
^{2}n^{2}, 10m^{2}n

Common factors: 2m^{2}n - 38a
^{3}b^{2}, 57ab^{2 }Common factors: 19ab^{2} - 11x
^{2}y^{3}, xy^{2 }Common factors: xy^{2} - 35p
^{2}q^{2}r, 40q^{3}r^{2}, 50pq^{2}r

Common factors: 5q^{2}r - 15x
^{3}y^{3}, 39x^{2}z^{2}, 48xy^{2}z^{3 }Common factors: 3x

**Exercise 78:**

**Solution 1:**

- 4a + 8b = 4 × a + 4 × 2 × b

= 4(a + 2b) - 5m + 15n = 5 × m + 5 × 3 × n

= 5(m + 3n) - abp – abq = ab × p – ab × q

= ab(p – q) - x
^{2 }+ x^{3}= x^{2}+ x^{2}× x

= x^{2}(1 + x) - mnx + mny = mn × x + mn × y

= mn(x + y) - 4x
^{2}y + 3xy^{2}= 4 × x × x × y + 3 × x × y × y

= xy(4x + 3y) - 15p
^{2}q – 20q = 3 × 5 × p × p × q – 4 × 5 × q

= 5q(3p^{2}– 4) - a
^{2}bc + abc^{2}= a × a × b × c + a × b × c × c

= abc(a + c) - 18m
^{2}n – 27m^{3}= 2 × 9 × m^{2}× n – 3 × 9 × m^{2}× m

= 9m^{2}(2n – 3m) - 24p
^{3}q^{2}+ 28p^{2}q^{3}= 4 × 6 × p^{2}× p × q^{2}+ 4 × 7 × p^{2}× q^{2}× q

= 4p^{2}q^{2}(6p + 7q)

**Exercise 79:**

**Solution 1:**

- ab + cd + ac + bd

= ab + ac + bd + cd

= a(b + c) + d(b + c)

= (a + d)(b + c) - 2x
^{2}+ 4x^{3}+ 2x + 1

= 4x^{3}+ 2x^{2}+ 2x + 1

= 2x^{2}(2x + 1) + 1(2x + 1)

= (2x + 1)(2x^{2}+ 1) - ax + bx – ay – by

= ax – ay + bx – by

= a(x – y) + b(x – y)

= (x – y)(a + b) - y – 1 + y
^{3}– y^{2 }= y – 1 + y^{3}– y^{2 }= 1(y – 1) + y^{2}(y – 1)

= (y – 1)(1 + y^{2}) - b
^{2}+ bc + ab + ac

= b^{2}+ bc + ab + ac

= b(b + c) + a(b + c)

= (b + c)(b + a) - 2x
^{2}+ xy – 2xy^{2}– y^{3 }= 2x^{2}– 2xy^{2}+ xy – y^{3 }= 2x(x – y^{2}) + y(x – y^{2})

= (x – y^{2})(2x + y) - 12pm + 18qm + 6pn + 9nq

= 12pm + 6pn + 18qm + 9nq

= 6p(2m + n) + 9q(2m + n)

= (2m + n)(6p + 9q)

= (2m + n)3(2p + 3q)

= 3(2p + 3q)(2m + 3n) - m
^{3}+ m^{2}+ m + 1

= m^{3}+ m^{2}+ m + 1

= m^{2}(m + 1) + 1(m + 1)

= (m^{2}+ 1)(m + 1) - am + an + al + bm + bl + bn

= am + an + al + bm + bl + bn

= a(m + n + l) + b(m + n + l)

= (a + b)(m + n + l) - 3y
^{3}– 6y^{2}+ 4y – 8

= 3y^{3}– 6y^{2}+ 4y – 8

= 3y^{2}(y – 2) + 4(y – 2)

= (3y^{2}+ 4)(y – 2)

**Exercise 80:**

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**Exercise 81:**

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