Question 1: Factorize by identifying common factors
(i) \( 6x + 12 \)
\[= 6(x + 2)\]
(ii) \( 15y^2 + 20y \)
\[= 5y(3y + 4)\]
(iii) \( -12x^2 – 3x \)
\[= -3x(4x + 1)\]
(iv) \( 4a^2b + 8ab^2 \)
\[= 4ab(a + 2b)\]
(v) \( xy – 3x^2 + 2x \)
\[= x(y – 3x + 2)\]
(vi) \( 3a^2b – 9ab^2 + 15ab \)
\[= 3ab(a – 3b + 5)\]
Question 2: Factorize and represent pictorially
(i) \( 5x + 15 \)
\[ 5x + 15 = 5(x + 3) \]
(ii) \( x^2 + 4x + 3 \)
\[ x^2 + 4x + 3 = x^2 + 3x + x + 3 \]
\[ = x(x + 3) + 1(x + 3) \]
\[ = (x + 3)(x + 1) \]
(iii) \( x^2 + 6x + 8 \)
\[ x^2 + 6x + 8 = x^2 + 4x + 2x + 8 \]
\[ = x(x + 4) + 2(x + 4) \]
\[ = (x + 4)(x + 2) \]
(iv) \( x^2 + 4x + 4 \)
\[ x^2 + 4x + 4 = x^2 + 2x + 2x + 4 \]
\[ = x(x + 2) + 2(x + 2) \]
\[ = (x + 2)(x + 2) \]
\[ = (x + 2)^2 \]
Question 3: Factorize
(i) \( x^2 + x – 12 \)
\[ x^2 + x – 12 = x^2 + 4x – 3x – 12 \]
\[ = x(x + 4) – 3(x + 4) \]
\[ = (x + 4)(x – 3) \]
(ii) \( x^2 + 7x + 10 \)
\[ x^2 + 7x + 10 = x^2 + 5x + 2x + 10 \]
\[ = x(x + 5) + 2(x + 5) \]
\[ = (x + 5)(x + 2) \]
(iii) \( x^2 – 6x + 8 \)
\[ x^2 – 6x + 8 = x^2 – 4x – 2x + 8 \]
\[ = x(x – 4) – 2(x – 4) \]
\[ = (x – 4)(x – 2) \]
(iv) \( x^2 – x – 56 \)
\[ x^2 – x – 56 = x^2 – 8x + 7x – 56 \]
\[ = x(x – 8) + 7(x – 8) \]
\[ = (x – 8)(x + 7) \]
(v) \( x^2 – 10x – 24 \)
\[ x^2 – 10x – 24 = x^2 – 12x + 2x – 24 \]
\[ = x(x – 12) + 2(x – 12) \]
\[ = (x – 12)(x + 2) \]
(vi) \( y^2 + 4y – 12 \)
\[ y^2 + 4y – 12 = y^2 + 6y – 2y – 12 \]
\[ = y(y + 6) – 2(y + 6) \]
\[ = (y + 6)(y – 2) \]
(vii) \( y^2 + 13y + 36 \)
\[ y^2 + 13y + 36 = y^2 + 9y + 4y + 36 \]
\[ = y(y + 9) + 4(y + 9) \]
\[ = (y + 9)(y + 4) \]
(viii) \( x^2 – x – 2 \)
\[ x^2 – x – 2 = x^2 – 2x + x – 2 \]
\[ = x(x – 2) + 1(x – 2) \]
\[ = (x – 2)(x + 1) \]
Question 4: Factorize
(i) \( 2x^2 + 7x + 3 \)
\[ 2x^2 + 7x + 3 = 2x^2 + 6x + x + 3 \]
\[ = 2x(x + 3) + 1(x + 3) \]
\[ = (2x + 1)(x + 3) \]
(ii) \( 2x^2 + 11x + 15 \)
\[2x^2 + 11x + 15 = 2x^2 + 6x + 5x + 15 \]
\[ = 2x(x + 3) + 5(x + 3) \]
\[ = (2x + 5)(x + 3) \]
(iii) \( 4x^2 + 13x + 3 \)
\[ 4x^2 + 13x + 3 = 4x^2 + 12x + x + 3 \]
\[ = 4x(x + 3) + 1(x + 3) \]
\[ = (4x + 1)(x + 3) \]
(iv) \( 3x^2 + 5x + 2 \)
\[ 3x^2 + 5x + 2 = 3x^2 + 3x + 2x + 2 \]
\[ = 3x(x + 1) + 2(x + 1) \]
\[ = (3x + 2)(x + 1) \]
(v) \( 3y^2 – 11y + 6 \)
\[ 3y^2 – 11y + 6 = 3y^2 – 9y – 2y + 6 \]
\[ = 3y(y – 3) – 2(y – 3) \]
\[ = (3y – 2)(y – 3) \]
(vi) \( 2y^2 – 5y + 2 \)
\[ 2y^2 – 5y + 2 = 2y^2 – 4y – y + 2 \]
\[ = 2y(y – 2) – 1(y – 2) \]
\[ = (2y – 1)(y – 2) \]
(vii) \( 4z^2 – 11z + 6 \)
\[ 4z^2 – 11z + 6 = 4z^2 – 8z – 3z + 6 \]
\[ = 4z(z – 2) – 3(z – 2) \]
\[ = (4z – 3)(z – 2) \]
(viii) \( 6 + 7x – 3x^2 \)
\[ 6 + 7x – 3x^2 = -3x^2 + 7x + 6 \]
\[ = -3x^2 + 9x – 2x + 6 \]
\[ = -3x(x – 3) – 2(x – 3) \]
\[ = (-3x – 2)(x – 3) \]
