RBSE Solutions Class 8 Maths Chapter 10 Factorization Exercise 10.2

Question 1. Factorize the following expressions.
(i) \( a^2 - 4 \)
(ii) \( a^2 - 49b^2 \)
(iii) \( p^3 - 121p \)
(iv) \( (a - b)^2 - c^2 \)
(v) \( a^4 - b^4 \)
(vi) \( 5x^3 - 125x \)
(vii) \( 63a^2 - 112b^2 \)
(viii) \( 9x^2y^2 - 16 \)
(ix) \( (l + m)^2 - (l - m)^2 \)
Answer:
(i) To factorize \( a^2 - 4 \), we use the difference of squares formula, \( x^2 - y^2 = (x - y)(x + y) \).
\( a^2 - 4 = a^2 - 2^2 \)
\( \implies (a - 2)(a + 2) \)
(ii) To factorize \( a^2 - 49b^2 \), we use the difference of squares formula.
\( a^2 - 49b^2 = a^2 - (7b)^2 \)
\( \implies (a - 7b)(a + 7b) \)
(iii) To factorize \( p^3 - 121p \), first take out the common factor \( p \).
\( p^3 - 121p = p(p^2 - 121) \)
Now, use the difference of squares formula for \( p^2 - 121 \), which is \( p^2 - 11^2 \).
\( \implies p(p - 11)(p + 11) \)
(iv) To factorize \( (a - b)^2 - c^2 \), we treat \( (a - b) \) as \( X \) and \( c \) as \( Y \) in the difference of squares formula \( X^2 - Y^2 = (X - Y)(X + Y) \).
\( (a - b)^2 - c^2 = [(a - b) - c][(a - b) + c] \)
\( \implies (a - b - c)(a - b + c) \)
(v) To factorize \( a^4 - b^4 \), we can write it as \( (a^2)^2 - (b^2)^2 \). This is a difference of squares.
\( a^4 - b^4 = (a^2)^2 - (b^2)^2 \)
\( \implies (a^2 - b^2)(a^2 + b^2) \)
Now, \( (a^2 - b^2) \) can be factored further using the difference of squares formula again.
\( \implies (a - b)(a + b)(a^2 + b^2) \)
(vi) To factorize \( 5x^3 - 125x \), first take out the common factor \( 5x \).
\( 5x^3 - 125x = 5x(x^2 - 25) \)
Now, use the difference of squares formula for \( x^2 - 25 \), which is \( x^2 - 5^2 \).
\( \implies 5x(x - 5)(x + 5) \)
(vii) To factorize \( 63a^2 - 112b^2 \), first find the greatest common factor (GCF). The GCF of 63 and 112 is 7.
\( 63a^2 - 112b^2 = 7(9a^2 - 16b^2) \)
Now, use the difference of squares formula for \( 9a^2 - 16b^2 \), which is \( (3a)^2 - (4b)^2 \).
\( \implies 7[(3a)^2 - (4b)^2] \)
\( \implies 7(3a - 4b)(3a + 4b) \)
(viii) To factorize \( 9x^2y^2 - 16 \), we can write it as \( (3xy)^2 - 4^2 \). This is a difference of squares.
\( 9x^2y^2 - 16 = (3xy)^2 - 4^2 \)
\( \implies (3xy - 4)(3xy + 4) \)
(ix) To factorize \( (l + m)^2 - (l - m)^2 \), we use the difference of squares formula where \( X = (l + m) \) and \( Y = (l - m) \).
\( (l + m)^2 - (l - m)^2 = [(l + m) - (l - m)][(l + m) + (l - m)] \)
\( \implies [l + m - l + m][l + m + l - m] \)
\( \implies [2m][2l] \)
\( \implies 4lm \)
In simple words: Factorizing means rewriting an expression as a product of simpler terms. For many of these, we use the "difference of squares" rule, where \( A^2 - B^2 \) becomes \( (A-B)(A+B) \). Sometimes, we first take out a common number or variable.

🎯 Exam Tip: Always look for common factors first before applying other factorization identities like difference of squares or grouping. This simplifies the expression and makes further steps easier.

Question 2. Factorize the following expressions.
(i) \( lx^2 + mx \)
(ii) \( 2x^3 + 2xy^2 + 2xz^2 \)
(iii) \( a(a + b) + 4(a + b) \)
(iv) \( (xy + y) + x + 1 \)
(v) \( 5a^2 - 15a - 6c + 2ac \)
(vi) \( am^2 + bm^2 + bn^2 + an^2 \)
(vii) \( y^2 + 2y - 48 \)
(viii) \( d^2 - 4d - 45 \)
(ix) \( m^2 + 16m + 63 \)
(x) \( n^2 - 19n - 92 \)
(xi) \( p^2 - 10p + 16 \)
(xii) \( x^2 + 4x - 45 \)
Answer:
(i) To factorize \( lx^2 + mx \), take out the common factor \( x \).
\( lx^2 + mx = x(lx + m) \)
(ii) To factorize \( 2x^3 + 2xy^2 + 2xz^2 \), take out the common factor \( 2x \).
\( 2x^3 + 2xy^2 + 2xz^2 = 2x(x^2 + y^2 + z^2) \)
(iii) To factorize \( a(a + b) + 4(a + b) \), take out the common factor \( (a + b) \).
\( a(a + b) + 4(a + b) = (a + b)(a + 4) \)
(iv) To factorize \( (xy + y) + x + 1 \), first group terms and find common factors.
\( (xy + y) + (x + 1) \)
\( \implies y(x + 1) + 1(x + 1) \)
Now, take out the common factor \( (x + 1) \).
\( \implies (x + 1)(y + 1) \)
(v) To factorize \( 5a^2 - 15a - 6c + 2ac \), group terms and find common factors.
\( 5a^2 - 15a + 2ac - 6c \)
\( \implies 5a(a - 3) + 2c(a - 3) \)
Now, take out the common factor \( (a - 3) \).
\( \implies (a - 3)(5a + 2c) \)
(vi) To factorize \( am^2 + bm^2 + bn^2 + an^2 \), group terms and find common factors.
\( am^2 + bm^2 + an^2 + bn^2 \)
\( \implies m^2(a + b) + n^2(a + b) \)
Now, take out the common factor \( (a + b) \).
\( \implies (a + b)(m^2 + n^2) \)
(vii) To factorize \( y^2 + 2y - 48 \), we need two numbers that multiply to -48 and add up to 2. These numbers are 8 and -6.
\( y^2 + 2y - 48 = y^2 + 8y - 6y - 48 \)
\( \implies y(y + 8) - 6(y + 8) \)
\( \implies (y + 8)(y - 6) \)
(viii) To factorize \( d^2 - 4d - 45 \), we need two numbers that multiply to -45 and add up to -4. These numbers are -9 and 5.
\( d^2 - 4d - 45 = d^2 - 9d + 5d - 45 \)
\( \implies d(d - 9) + 5(d - 9) \)
\( \implies (d - 9)(d + 5) \)
(ix) To factorize \( m^2 + 16m + 63 \), we need two numbers that multiply to 63 and add up to 16. These numbers are 9 and 7.
\( m^2 + 16m + 63 = m^2 + 9m + 7m + 63 \)
\( \implies m(m + 9) + 7(m + 9) \)
\( \implies (m + 9)(m + 7) \)
(x) To factorize \( n^2 - 19n - 92 \), we need two numbers that multiply to -92 and add up to -19. These numbers are -23 and 4.
\( n^2 - 19n - 92 = n^2 - 23n + 4n - 92 \)
\( \implies n(n - 23) + 4(n - 23) \)
\( \implies (n - 23)(n + 4) \)
(xi) To factorize \( p^2 - 10p + 16 \), we need two numbers that multiply to 16 and add up to -10. These numbers are -8 and -2.
\( p^2 - 10p + 16 = p^2 - 8p - 2p + 16 \)
\( \implies p(p - 8) - 2(p - 8) \)
\( \implies (p - 8)(p - 2) \)
(xii) To factorize \( x^2 + 4x - 45 \), we need two numbers that multiply to -45 and add up to 4. These numbers are 9 and -5.
\( x^2 + 4x - 45 = x^2 + 9x - 5x - 45 \)
\( \implies x(x + 9) - 5(x + 9) \)
\( \implies (x + 9)(x - 5) \)
In simple words: When factorizing, we often look for common terms to pull out or try to find pairs of numbers that fit specific multiplication and addition rules. This helps break down complex expressions into simpler parts, making them easier to work with.

🎯 Exam Tip: For quadratic expressions like \( x^2 + bx + c \), always look for two numbers that multiply to \( c \) and add up to \( b \). This is a key method for factorization.