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Mathematics Question on What is Factorisation?

Factorise the following expressions

  1. 7x−427x - 42
  2. 6p−12q6p - 12q
  3. 7a2+14a7a ^2 + 14a
  4. −16z+20z3-16z + 20z ^3
  5. 20l2m+30 alm20l^ 2m + 30\, alm
  6. 5x2y−15xy25x^ 2 y - 15xy^2
  7. 10a2−15b2+20c210a^ 2 - 15b^ 2 + 20c^ 2
  8. −4a2+4ab−4ca-4a^ 2 + 4ab - 4 ca
  9. x2yz+xy2z+xyz2x ^2 yz + xy^2 z + xyz^2
  10. ax2y+bxy2+cxyzax^2 y + bxy^2 + cxyz
Answer

(i) 7x=7×x7x = 7 \times x
42=2×3×742 = 2 \times 3 \times 7
The common factor is 77.
∴ 7x−42=(7×x)−(2×3×7)=7(x−6)7x - 42 = (7 \times x) - (2 \times 3 \times 7) = 7 (x - 6)

(ii) 6p=2×3×p6p = 2 \times 3 \times p
12q=2×2×3×q12q = 2 \times 2 \times 3 \times q
The common factors are 22 and 33.
∴ 6p−12q=(2×3×p)−(2×2×3×q)6p - 12q = (2 \times 3 \times p) - (2 \times 2 \times 3 \times q)
= 2×3[p−(2×q)]2 \times 3 [p - (2 \times q)]
= 6(p−2q)6 (p - 2q)

(iii) 7a2=7×a×a7a^2= 7 \times a \times a
14a=2×7×a14a = 2 \times 7 \times a
The common factors are 77 and aa.
∴ 7a2+14a=(7×a×a)+(2×7×a)7a^2+ 14a = (7 \times a \times a) + (2 \times 7 \times a)
= 7×a[a+2]=7a(a+2)7 \times a [a + 2] = 7a (a + 2)

(iv) 16z=2×2×2×2×z16z = 2 \times 2 \times 2 \times 2 \times z \\\\
20z3=2×2×5×z×z×z20z^3= 2 \times 2 \times 5 \times z \times z \times z
The common factors are 22, 22, and zz.
∴ −16z+20z3=−(2×2×2×2×z)+(2×2×5×z×z×z)-16z + 20z^3 = - (2 \times 2 \times 2 \times 2 \times z) + (2 \times 2 \times 5 \times z \times z \times z)
= (2×2×z)[−(2×2)+(5×z×z)](2 \times 2 \times z) [- (2 \times 2) + (5 \times z \times z)]
= 4z(−4+5z2)4z (- 4 + 5z^2)

(v) 20l2m=2×2×5×l×l×m20l^2m = 2 \times 2 \times 5 \times l \times l \times m
30alm=2×3×5×a×l×m30alm = 2 \times 3 \times 5 \times a \times l \times m
The common factors are 2,52, 5, ll, and mm.
∴ 20l2m+30alm=(2×2×5×l×l×m)+(2×3×5×a×l×m)20l^2m + 30alm = (2 \times 2 \times 5 \times l \times l \times m) + (2 \times 3 \times 5 \times a \times l \times m)
= (2×5×l×m)[(2×l)+(3×a)](2 \times 5 \times l \times m) [(2 \times l) + (3 \times a)]
= 10lm  (2l+3a)10lm\; (2l + 3a)

(vi) 5x2y=5×x×x×y5x^2y = 5 \times x \times x \times y
15xy2=3×5×x×y×y15xy^2= 3 \times 5 \times x \times y \times y
The common factors are 55, xx, and yy.
∴ 5x2y−15xy25x^2y - 15xy^2
= (5×x×x×y)−(3×5×x×y×y)(5 \times x \times x \times y) - (3 \times 5 \times x \times y \times y)
= 5×x×y[x−(3×y)]5 \times x \times y [x - (3 \times y)]
= 5xy(x−3y)5xy (x - 3y)

(vii) 10a2=2×5×a×a10a^2= 2 \times 5 \times a \times a
15b2=3×5×b×b15b^2= 3 \times 5 \times b \times b
20c2=2×2×5×c×c20c^2= 2 \times 2 \times 5 \times c \times c
The common factor is 55.
10a2−15b2+20c210a^2- 15b^2+ 20c^2
= (2×5×a×a)−(3×5×b×b)+(2×2×5×c×c)(2 \times 5 \times a \times a) - (3 \times 5 \times b \times b) + (2 \times 2 \times 5 \times c \times c)
= 5[(2×a×a)−(3×b×b)+(2×2×c×c)]5 [(2 \times a \times a) - (3 \times b \times b) + (2 \times 2 \times c \times c)]
= 5(2a2−3b2+4c2)5 (2a^2- 3b^2+ 4c^2)

(viii) 4a2=2×2×a×a4a^2= 2 \times 2 \times a \times a
4ab=2×2×a×b4ab = 2 \times 2 \times a \times b
4ca=2×2×c×a4ca = 2 \times 2 \times c \times a
The common factors are 2,2,2, 2, and aa.
∴ −4a2+4ab−4ca=−(2×2×a×a)+(2×2×a×b)−(2×2×c×a)-4a^2+ 4ab - 4ca = - (2 \times 2 \times a \times a) + (2 \times 2 \times a \times b) - (2 \times 2 \times c \times a)
=2×2×a[−(a)+b−c] 2 \times 2 \times a [- (a) + b - c]
= 4a(−a+b−c)4a (-a + b - c)

(ix) x2yz=x×x×y×zx^2yz = x \times x \times y \times z
xy2z=x×y×y×zxy^2z = x \times y \times y \times z
xyz2=x×y×z×zxyz^2 = x \times y \times z \times z
The common factors are x,yx, y, and zz.
∴ x2yz+xy2z+xyz2x^2yz + xy^2z + xyz^2
= (x×x×y×z)+(x×y×y×z)+(x×y×z×z)(x \times x \times y \times z) + (x \times y \times y \times z) + (x \times y \times z \times z)
= x×y×z[x+y+z]x \times y \times z [x + y + z]
= xyz(x+y+z)xyz (x + y + z)

(x) ax2y=a×x×x×yax^2y = a \times x \times x \times y
bxy2=b×x×y×ybxy^2 = b \times x \times y \times y
cxyz=c×x×y×zcxyz = c \times x \times y \times z
The common factors are xx and yy.