Division of Algebraic Expressions: This involves dividing one algebraic expression by another. It’s similar to dividing numbers, but with variables included.
Division of a monomial by another monomial: Take the coefficients and divide them as usual. Then, divide the variables by subtracting their exponents. For example, if you have 8x³ divided by 2x, divide 8 by 2 to get 4, and subtract the exponents of x (3-1) to get x². So, 8x³ divided by 2x is 4x².
Division of a polynomial by a monomial: Divide each term in the polynomial by the monomial separately. Let’s say we have 15x²y + 10xy² divided by 5xy. Divide 15x²y by 5xy to get 3x, and 10xy² by 5xy to get 2y. So, the answer is 3x + 2y.
Division of Algebraic Expressions Continued (Polynomial ÷ Polynomial): This is like long division but with polynomials. For example, to divide x² + 3x + 2 by x + 1, we see how many times x fits into x² (which is x times), then multiply the divisor (x + 1) by this quotient (x) and subtract it from the dividend, and bring down the next term to repeat the process. The result will be x + 2.
NCERT Solutions for Class 8 Maths Exercise 12.3 Chapter 12 Factorisation
1. Carry out the following divisions.
(i) 28x⁴ ÷ 56x (ii) –36y³ ÷ 9y² (iii) 66pq²r³ ÷ 11qr²
(iv) 34x³y³z³ ÷ 51xy²z³ (v) 12a⁸b⁸ ÷ (– 6a⁶b⁴)
(i) 28x⁴ ÷ 56x
28x⁴ = 2 × 2 × 7 × x × x × x × x
56x = 2 × 2 × 2 × 7 × x
28x⁴/56x = (2 × 2 × 7 × x × x × x × x)/(2 × 2 × 2 × 7)
= (x × x × x)/2
= 1/2 x³
Method 2:
= 28x⁴ / 56x
= (28 / 56) × (x⁴ / x)
= (1/2)x³
(ii) –36y³ ÷ 9y²
–36y³ = –(2 × 2 × 3 × 3) × y × y × y
9y² = 3 × 3 × y × y
–36y³ / 9y² = (–(2 × 2 × 3 × 3) × y × y × y) / (3 × 3 × y × y)
= –4y
Method 2:
= –36y³ / 9y²
= (–36 / 9) × (y³ / y²)
= –4y
(iii) 66pq²r³ ÷ 11qr²
66pq²r³ = 2 × 3 × 11 × p × q × q × r × r × r
11qr² = 11 × q × r × r
66pq²r³ / 11qr² = (2 × 3 × 11 × p × q × q × r × r × r) / (11 × q × r × r)
= 6pqr
Method 2:
= 66pq²r³ / 11qr²
= (66 / 11) × (pq²r³ / qr²)
= 6pqr
(iv) 34x³y³z³ ÷ 51xy²z³
34x³y³z³ = 2 × 17 × x × x × x × y × y × y × z × z × z
51xy²z³ = 3 × 17 × x × y × y × z × z × z
34x³y³z³ / 51xy²z³ = (2 × 17 × x × x × x × y × y × y × z × z × z) / (3 × 17 × x × y × y × z × z × z)
= 2/3 x²y
Method 2:
= 34x³y³z³ / 51xy²z³
= (34 / 51) × (x³y³z³ / xy²z³)
= 2/3 x²y
(v) 12a⁸b⁸ ÷ (– 6a⁶b⁴)
12a⁸b⁸ = 2 × 2 × 3 × a × a × a × a × a × a × a × a × b × b × b × b × b × b × b × b
–6a⁶b⁴ = –(2 × 3) × a × a × a × a × a × a × b × b × b × b
12a⁸b⁸ / (– 6a⁶b⁴) = (2 × 2 × 3 × a × a × a × a × a × a × a × a × b × b × b × b × b × b × b × b) / (–(2 × 3) × a × a × a × a × a × a × b × b × b × b)
= –2a²b⁴
Method 2:
= 12a⁸b⁸ / (– 6a⁶b⁴)
= (12 / –6) × (a⁸b⁸ / a⁶b⁴)
= –2a²b⁴
2. Divide the given polynomial by the given monomial.
(i) (5x² – 6x) ÷ 3x (ii) (3y⁸ – 4y⁶ + 5y⁴) ÷ y⁴
(iii) 8(x³y²z² + x²y3z² + x²y²z³) ÷ 4x²y²z² (iv) (x³ + 2x² + 3x) ÷ 2x
(v) (p³q⁶ – p⁶q³) ÷ p³q³
(i) (5x² – 6x) ÷ 3x
= (5x² / 3x) – (6x / 3x)
= 5/3 x – 2
= (5x – 6)/3 (taking LCM as 3)
= 1/3 (5x – 6)(
(ii) (3y⁸ – 4y⁶ + 5y⁴) ÷ y⁴
= (3y⁸ / y⁴) – (4y⁶ / y⁴) + (5y⁴ / y⁴)
= 3y⁴ – 4y² + 5
(iii) 8(x³y²z² + x²y³z² + x²y²z³) ÷ 4x²y²z²
= 8(x³y²z² / 4x²y²z²) + 8(x²y³z² / 4x²y²z²) + 8(x²y²z³ / 4x²y²z²)
= 2xz² + 2yz² + 2z
(iv) (x³ + 2x² + 3x) ÷ 2x
= (x³ / 2x) + (2x² / 2x) + (3x / 2x)
= 1/2 x² + x + 3/2
= 1/2 (x² + 2x + 3)
(v) (p³q⁶ – p⁶q³) ÷ p³q³
= (p³q⁶ / p³q³) – (p⁶q³ / p³q³)
= q³ – p³
3. Work out the following divisions.
(i) (10x – 25) ÷ 5 (ii) (10x – 25) ÷ (2x – 5)
(iii) 10y(6y + 21) ÷ 5(2y + 7) (iv) 9x²y²(3z – 24) ÷ 27xy(z – 8)
(v) 96abc(3a – 12) (5b – 30) ÷ 144(a – 4) (b – 6)
(i) (10x – 25) ÷ 5
First, simplify the numerator:
10x – 25 can be factored as 5(2x – 5)
Now, divide by 5:
= 5(2x – 5) / 5
Simplifying:
= 2x – 5
(ii) (10x – 25) ÷ (2x – 5)
First, factor the numerator:
10x – 25 = 5(2x – 5)
The expression simplifies to:
= 5(2x – 5) / (2x – 5)
= 5
(iii) 10y(6y + 21) ÷ 5(2y + 7)
First, factor out common terms in the numerator:
10y(6y + 21) = 10y × 3(2y + 7)
The expression simplifies to:
= 10y × 3(2y + 7) / 5(2y + 7)
Since the terms (2y + 7) cancel out:
= 10y × 3 / 5
= 6y
(iv) 9x²y²(3z – 24) ÷ 27xy(z – 8)
First, simplify the numerator and denominator:
9x²y²(3z – 24) = 9x²y² × 3(z – 8)
The expression simplifies to:
= 9x²y² × 3(z – 8) / 27xy(z – 8)
Cancel out the common terms (z – 8) and simplify coefficients:
= (9 × 3)x²y² / 27xy
= xy
(v) 96abc(3a – 12) (5b – 30) ÷ 144(a – 4) (b – 6)
First, factor out the common terms in the expressions and simplify:
96abc(3a – 12)(5b – 30) simplifies to 96abc × 3(a – 4) × 5(b – 6)
= 480abc(a – 4)(b – 6)
The denominator simplifies to:
144(a – 4)(b – 6)
Now, simplifying the original expression:
= 480abc(a – 4)(b – 6) / 144(a – 4)(b – 6)
Simplifying the coefficients and cancelling out common terms correctly:
= 10abc
4. Divide as directed.
(i) 5(2x + 1) (3x + 5) ÷ (2x + 1) (ii) 26xy(x + 5) (y – 4) ÷ 13x(y – 4)
(iii) 52pqr (p + q) (q + r) (r + p) ÷ 104pq(q + r) (r + p)
(iv) 20(y + 4) (y2 + 5y + 3) ÷ 5(y + 4) (v) x(x + 1) (x + 2) (x + 3) ÷ x(x + 1)
(i) 5(2x + 1) (3x + 5) ÷ (2x + 1)
= 5 × (3x + 5) (as the term (2x + 1) cancels out)
= 15x + 25
(ii) 26xy(x + 5) (y – 4) ÷ 13x(y – 4)
= 26xy × (x + 5) / 13x (as the term (y – 4) cancels out)
= 2y × (x + 5)
= 2yx + 10y
(iii) 52pqr(p + q) (q + r) (r + p) ÷ 104pq(q + r) (r + p)
= 52pqr × (p + q) / 104pq (as the terms (q + r) and (r + p) cancel out)
= 1/2 × r × (p + q)
= 1/2 rp + 1/2 rq
(iv) 20(y + 4) (y² + 5y + 3) ÷ 5(y + 4)
= 20 × (y² + 5y + 3) / 5 (as the term (y + 4) cancels out)
= 4y² + 20y + 12
(v) x(x + 1) (x + 2) (x + 3) ÷ x(x + 1)
= (x + 2) × (x + 3) (as the terms x and (x + 1) cancel out)
= x² + 5x + 6
5. Factorise the expressions and divide them as directed.
(i) (y² + 7y + 10) ÷ (y + 5) (ii) (m² – 14m – 32) ÷ (m + 2)
(iii) (5p² – 25p + 20) ÷ (p – 1) (iv) 4yz(z² + 6z – 16) ÷ 2y(z + 8)
(v) 5pq(p² – q²) ÷ 2p(p + q)
(vi) 12xy(9x² – 16y²) ÷ 4xy(3x + 4y) (vii) 39y³(50y² – 98) ÷ 26y²(5y + 7)
(i) (y² + 7y + 10) ÷ (y + 5)
Step 1: Factorise y² + 7y + 10
= y² + 5y + 2y + 10
= y(y + 5) + 2(y + 5)
= (y + 2)(y + 5)
Step 2: Divide by (y + 5)
= (y + 2)(y + 5) ÷ (y + 5)
= y + 2
(ii) (m² – 14m – 32) ÷ (m + 2)
Step 1: Factorise m² – 14m – 32
= m² – 16m + 2m – 32
= m(m – 16) + 2(m – 16)
= (m – 16)(m + 2)
Step 2: Divide by (m + 2)
= (m – 16)(m + 2) ÷ (m + 2)
= m – 16
(iii) (5p² – 25p + 20) ÷ (p – 1)
Step 1: Factorise 5p² – 25p + 20
= 5p² – 5p – 20p + 20
= 5p(p – 1) – 20(p – 1)
= 5(p – 1)(p – 4)
Step 2: Divide by (p – 1)
= 5(p – 1)(p – 4) ÷ (p – 1)
= 5(p – 4)
(iv) 4yz(z² + 6z – 16) ÷ 2y(z + 8)
Step 1: Factorise z² + 6z – 16
= z² + 8z – 2z – 16
= z(z + 8) – 2(z + 8)
= (z + 8)(z – 2)
Step 2: Multiply by 4yz and divide by 2y(z + 8)
= 4yz(z + 8)(z – 2) ÷ 2y(z + 8)
= 2z(z – 2)
(v) 5pq(p² – q²) ÷ 2p(p + q)
Step 1: Factorise p² – q² (Difference of two squares)
= (p + q)(p – q)
Step 2: Multiply by 5pq and divide by 2p(p + q)
= 5pq(p + q)(p – q) ÷ 2p(p + q)
= 5/2 × q(p – q)
(vi) 12xy(9x² – 16y²) ÷ 4xy(3x + 4y)
Step 1: Factorise 9x² – 16y² (Difference of two squares)
= (3x + 4y)(3x – 4y)
Step 2: Multiply by 12xy and divide by 4xy(3x + 4y)
= 12xy(3x + 4y)(3x – 4y) ÷ 4xy(3x + 4y)
= 3(3x – 4y)
(vii) 39y³(50y² – 98) ÷ 26y²(5y + 7)
Step 1: Factorise 50y² – 98
= 2 × 25y² – 2 × 49
= 2(25y² – 49)
= 2(5y + 7)(5y – 7)
Step 2: Multiply by 39y³ and divide by 26y²(5y + 7)
= 39y³ × 2(5y + 7)(5y – 7) ÷ 26y²(5y + 7)
= 3y(5y – 7)
