GSEB Class 8 Maths Solutions Chapter 14 Factorization Exercise 14.1

Question 1. Find the common factors of the given terms?
1. 12x, 36
2. 2y, 22y
3. 14pq, 28p²q²
4. 2x, 3x², 4
5. 6abc, 24ab², 12a²b
6. 10pq, 20qr, 30rp
7. 3x²y³, 10x³y², 6x²y²z
Answer:
1. To find the common factors of \( 12x \) and \( 36 \):
\( 12x = 2 \times 2 \times 3 \times x \)
\( 36 = 2 \times 2 \times 3 \times 3 \)
The common factor is \( 2 \times 2 \times 3 = 12 \).
2. To find the common factors of \( 2y \) and \( 22y \):
\( 2y = 2 \times y \)
\( 22y = 2 \times 11 \times y \)
The common factor is \( 2 \times y = 2y \).
3. To find the common factors of \( 14pq \) and \( 28p^2q^2 \):
\( 14pq = 2 \times 7 \times p \times q \)
\( 28p^2q^2 = 2 \times 2 \times 7 \times p \times p \times q \times q \)
The common factor is \( 2 \times 7 \times p \times q = 14pq \).
4. To find the common factors of \( 2x \), \( 3x^2 \), and \( 4 \):
\( 2x = 1 \times 2 \times x \)
\( 3x^2 = 1 \times 3 \times x \times x \)
\( 4 = 1 \times 2 \times 2 \)
The common factor is \( 1 \). (Note: \( 1 \) is a factor of every term.)
5. To find the common factors of \( 6abc \), \( 24ab^2 \), and \( 12a^2b \):
\( 6abc = 2 \times 3 \times a \times b \times c \)
\( 24ab^2 = 2 \times 2 \times 2 \times 3 \times a \times b \times b \)
\( 12a^2b = 2 \times 2 \times 3 \times a \times a \times b \)
The common factor is \( 2 \times 3 \times a \times b = 6ab \).
6. To find the common factors of \( 10pq \), \( 20qr \), and \( 30rp \):
\( 10pq = 2 \times 5 \times p \times q \)
\( 20qr = 2 \times 2 \times 5 \times q \times r \)
\( 30rp = 2 \times 3 \times 5 \times r \times p \)
The common factor is \( 2 \times 5 = 10 \).
7. To find the common factors of \( 3x^2y^3 \), \( 10x^3y^2 \), and \( 6x^2y^2z \):
\( 3x^2y^3 = 3 \times x \times x \times y \times y \times y \)
\( 10x^3y^2 = 2 \times 5 \times x \times x \times x \times y \times y \)
\( 6x^2y^2z = 2 \times 3 \times x \times x \times y \times y \times z \)
The common factor is \( x \times x \times y \times y = x^2y^2 \).
In simple words: To find the factors that are shared between terms, break each term down into its simplest parts. Then, pick out all the parts that show up in every single term. Multiply those common parts together to get your final common factor.

Exam Tip: Always make sure you break down each term completely into its prime factors to accurately identify the greatest common factor (GCF).

Question 2. Factorise the following expressions:
1. 7x-42
2. 6p - 12q
3. 7a² + 14a
4. -16z + 20z³
5. 20l²m + 30alm
6. 5x²y – 15xy²
7. 10 a² - 15 b² + 20 c²
8. -4a² + 4ab – 4ca
9. x²yz + xy²z + xyz²
10. ax²y + bxy² + cxyz
Answer:
1. Factorise \( 7x - 42 \):
\( 7x = 7 \times x \)
\( 42 = 7 \times 6 \)
\( \therefore 7x - 42 = 7(x) - 7(6) = 7(x - 6) \)
2. Factorise \( 6p - 12q \):
\( 6p = 6 \times p \)
\( 12q = 6 \times 2 \times q \)
\( \therefore 6p - 12q = 6(p) - 6(2q) = 6(p - 2q) \)
3. Factorise \( 7a^2 + 14a \):
\( 7a^2 = 7a \times a \)
\( 14a = 7a \times 2 \)
\( \therefore 7a^2 + 14a = 7a(a) + 7a(2) = 7a(a + 2) \)
4. Factorise \( -16z + 20z^3 \):
\( -16z = 4z \times (-4) \)
\( 20z^3 = 4z \times (5z^2) \)
\( \therefore -16z + 20z^3 = 4z(-4) + 4z(5z^2) = 4z(-4 + 5z^2) \)
5. Factorise \( 20l^2m + 30alm \):
\( 20l^2m = 10lm \times 2l \)
\( 30alm = 10lm \times 3a \)
\( \therefore 20l^2m + 30alm = 10lm(2l) + 10lm(3a) = 10lm(2l + 3a) \)
6. Factorise \( 5x^2y - 15xy^2 \):
\( 5x^2y = 5xy \times x \)
\( 15xy^2 = 5xy \times 3y \)
\( \therefore 5x^2y - 15xy^2 = 5xy(x) - 5xy(3y) = 5xy(x - 3y) \)
7. Factorise \( 10a^2 - 15b^2 + 20c^2 \):
\( 10a^2 = 5 \times 2a^2 \)
\( -15b^2 = 5 \times (-3b^2) \)
\( 20c^2 = 5 \times 4c^2 \)
\( \therefore 10a^2 - 15b^2 + 20c^2 = 5(2a^2 - 3b^2 + 4c^2) \)
8. Factorise \( -4a^2 + 4ab - 4ca \):
\( -4a^2 = 4a \times (-a) \)
\( 4ab = 4a \times b \)
\( -4ca = 4a \times (-c) \)
\( \therefore -4a^2 + 4ab - 4ca = 4a(-a) + 4a(b) + 4a(-c) = 4a(-a + b - c) \)
9. Factorise \( x^2yz + xy^2z + xyz^2 \):
\( x^2yz = xyz \times x \)
\( xy^2z = xyz \times y \)
\( xyz^2 = xyz \times z \)
\( \therefore x^2yz + xy^2z + xyz^2 = xyz(x) + xyz(y) + xyz(z) = xyz(x + y + z) \)
10. Factorise \( ax^2y + bxy^2 + cxyz \):
\( ax^2y = xy \times ax \)
\( bxy^2 = xy \times by \)
\( cxyz = xy \times cz \)
\( \therefore ax^2y + bxy^2 + cxyz = xy(ax) + xy(by) + xy(cz) = xy(ax + by + cz) \)
In simple words: To factorize these expressions, identify the largest common factor shared by all terms. Then, write the common factor outside a bracket and place the remaining parts inside the bracket. For four-term expressions, try grouping them into two pairs first.

Exam Tip: Always double-check your factorization by multiplying the terms back out. This helps ensure that your factored expression is equivalent to the original one.

Question 3. Factorise:
1. x² + xy + 8x + 8y
2. 15xy – 6x + 5y - 2
3. ax + bx – ay – by
4. 15pq + 15 + 9q + 25p
5. z − 7 + 7xy – xyz
Answer:
1. Factorise \( x^2 + xy + 8x + 8y \):
Group the terms: \( (x^2 + xy) + (8x + 8y) \)
Factor out common terms from each group: \( x(x + y) + 8(x + y) \)
Factor out the common binomial: \( (x + y)(x + 8) \)
2. Factorise \( 15xy - 6x + 5y - 2 \):
Group the terms: \( (15xy - 6x) + (5y - 2) \)
Factor out common terms from each group: \( 3x(5y - 2) + 1(5y - 2) \)
Factor out the common binomial: \( (5y - 2)(3x + 1) \)
3. Factorise \( ax + bx - ay - by \):
Group the terms: \( (ax + bx) + (-ay - by) \)
Factor out common terms from each group: \( x(a + b) - y(a + b) \)
Factor out the common binomial: \( (a + b)(x - y) \)
4. Factorise \( 15pq + 15 + 9q + 25p \):
Regroup the terms: \( 15pq + 9q + 25p + 15 \)
Group the terms: \( (15pq + 9q) + (25p + 15) \)
Factor out common terms from each group: \( 3q(5p + 3) + 5(5p + 3) \)
Factor out the common binomial: \( (5p + 3)(3q + 5) \)
5. Factorise \( z - 7 + 7xy - xyz \):
Regroup the terms: \( (z - 7) + (7xy - xyz) \)
Factor out common terms from each group: \( 1(z - 7) + xy(7 - z) \)
To make \( (7 - z) \) become \( (z - 7) \), factor out \( -1 \) from \( xy(7 - z) \):
\( 1(z - 7) - xy(z - 7) \)
Factor out the common binomial: \( (z - 7)(1 - xy) \)
In simple words: For these types of problems, try grouping the terms in pairs. Find what's common in each pair, then see if there's a common bracket you can take out again. Sometimes you might need to rearrange the terms or factor out a negative sign to create a common bracket.

Exam Tip: When factorising by grouping, remember that the goal is to make a common bracket appear. If your initial grouping doesn't work, try rearranging the terms and grouping them differently.