Factorise.

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(i) ax² + bx

Here common factor is x

ax² + bx = x(ax + b)

(ii) 7p² + 21q²

Here common factor is 7

7p² + 21q² = 7(p² + 3q²)

(iii) 2x³ + 2xy² + 2xz²

Here common factor is 2x

2x³ + 2xy² + 2xz² = 2x(x² + y² + z²)

(iv) am² + bm² + bn² + an²

Here common factor is m² in first two terms and n² in the last two terms

am² + bm² + bn² + an² = m²(a + b) + n²(a + b)

am² + bm² + bn² + an² = (m² + n²) + (a + b)

(v) (lm + l) + m + 1

On opening the bracket, we get

lm + l + m + 1

Here common factor is l in first two terms and 1 in the last two terms

lm + l + m + 1 = l(m + 1) + 1(m + 1)

lm + l + m + 1 = (l + 1) + (m + 1)

(vi) y(y + z) + 9(y + z)

In the brackets y + z is common,

(y + z) (9 + y) = (y + z)(y + 9)

(vii) 5y² - 20y – 8z + 2yz

Taking 5y common in first two pairs and 2z in the last two terms

5y(y - 4)-2z(y - 4) = (y - 4)(5y – 2z)

(viii) 10ab + 4a + 5b + 2

Taking 2a common in first two pairs and 1 in the last two terms

10ab + 4a + 5b + 2 = 2a(5b + 2) + 1(5b + 2)

10ab + 4a + 5b + 2 = (5b + 2) + (2a + 1)

(ix) _{6xy – 4y + 6 – 9x}

Taking 2y common in first two pairs and -3 in the last two terms

6xy – 4y + 6 – 9x = 2y(3x - 2) - 3(3x - 2)

6xy – 4y + 6 – 9x = (3x - 2) + (2y - 3)