Let's factorize the given expressions step by step:
### (i) \(x^2 + 3x - 40\)
1. Identify \(A = 1\), \(B = 3\), and \(C = -40\).
2. Calculate \(A \cdot C = 1 \cdot (-40) = -40\).
3. Find two numbers that multiply to \(-40\) and add to \(3\). The numbers are \(8\) and \(-5\).
4. Rewrite \(3x\) as \(8x - 5x\):
\[
x^2 + 8x - 5x - 40
\]
5. Factor by grouping:
\[
x(x + 8) - 5(x + 8)
\]
6. Combine the factors:
\[
(x - 5)(x + 8)
\]
### (ii) \(x^2 - 3x - 40\)
1. Identify \(A = 1\), \(B = -3\), and \(C = -40\).
2. Calculate \(A \cdot C = 1 \cdot (-40) = -40\).
3. Find two numbers that multiply to \(-40\) and add to \(-3\). The numbers are \(5\) and \(-8\).
4. Rewrite \(-3x\) as \(5x - 8x\):
\[
x^2 + 5x - 8x - 40
\]
5. Factor by grouping:
\[
x(x + 5) - 8(x + 5)
\]
6. Combine the factors:
\[
(x - 8)(x + 5)
\]
### (iii) \(x^2 + 5x - 14\)
1. Identify \(A = 1\), \(B = 5\), and \(C = -14\).
2. Calculate \(A \cdot C = 1 \cdot (-14) = -14\).
3. Find two numbers that multiply to \(-14\) and add to \(5\). The numbers are \(7\) and \(-2\).
4. Rewrite \(5x\) as \(7x - 2x\):
\[
x^2 + 7x - 2x - 14
\]
5. Factor by grouping:
\[
x(x + 7) - 2(x + 7)
\]
6. Combine the factors:
\[
(x - 2)(x + 7)
\]
### (iv) \(x^2 - 3x - 4\)
1. Identify \(A = 1\), \(B = -3\), and \(C = -4\).
2. Calculate \(A \cdot C = 1 \cdot (-4) = -4\).
3. Find two numbers that multiply to \(-4\) and add to \(-3\). The numbers are \(-4\) and \(1\).
4. Rewrite \(-3x\) as \(-4x + 1x\):
\[
x^2 - 4x + 1x - 4
\]
5. Factor by grouping:
\[
x(x - 4) + 1(x - 4)
\]
6. Combine the factors:
\[
(x + 1)(x - 4)
\]
### (v) \(x^2 - 3x - 40\)
This is the same as (ii) and has already been factored:
\[
(x - 8)(x + 5)
\]
### (vi) \(3x^2 - 10x + 8\)
1. Identify \(A = 3\), \(B = -10\), and \(C = 8\).
2. Calculate \(A \cdot C = 3 \cdot 8 = 24\).
3. Find two numbers that multiply to \(24\) and add to \(-10\). The numbers are \(-6\) and \(-4\).
4. Rewrite \(-10x\) as \(-6x - 4x\):
\[
3x^2 - 6x - 4x + 8
\]
5. Factor by grouping:
\[
3x(x - 2) - 4(x - 2)
\]
6. Combine the factors:
\[
(3x - 4)(x - 2)
\]
### (vii) \(12x^2 - x - 35\)
1. Identify \(A = 12\), \(B = -1\), and \(C = -35\).
2. Calculate \(A \cdot C = 12 \cdot (-35) = -420\).
3. Find two numbers that multiply to \(-420\) and add to \(-1\). The numbers are \(20\) and \(-21\).
4. Rewrite \(-x\) as \(20x - 21x\):
\[
12x^2 + 20x - 21x - 35
\]
5. Factor by grouping:
\[
4x(3x + 5) - 7(3x + 5)
\]
6. Combine the factors:
\[
(4x - 7)(3x + 5)
\]
### (viii) \(3x^2 - 5x - 2\)
1. Identify \(A = 3\), \(B = -5\), and \(C = -2\).
2. Calculate \(A \cdot C = 3 \cdot (-2) = -6\).
3. Find two numbers that multiply to \(-6\) and add to \(-5\). The numbers are \(-6\) and \(1\).
4. Rewrite \(-5x\) as \(-6x + x\):
\[
3x^2 - 6x + x - 2
\]
5. Factor by grouping:
\[
3x(x - 2) + 1(x - 2)
\]
6. Combine the factors:
\[
(3x + 1)(x - 2)
\]
### (ix) \(3x^2 - 7x + 4\)
1. Identify \(A = 3\), \(B = -7\), and \(C = 4\).
2. Calculate \(A \cdot C = 3 \cdot 4 = 12\).
3. Find two numbers that multiply to \(12\) and add to \(-7\). The numbers are \(-3\) and \(-4\).
4. Rewrite \(-7x\) as \(-3x - 4x\):
\[
3x^2 - 3x - 4x + 4
\]
5. Factor by grouping:
\[
3x(x - 1) - 4(x - 1)
\]
6. Combine the factors:
\[
(3x - 4)(x - 1)
\]
### (x) \(7x^2 - 8x + 1\)
1. Identify \(A = 7\), \(B = -8\), and \(C = 1\).
2. Calculate \(A \cdot C = 7 \cdot 1 = 7\).
3. Find two numbers that multiply to \(7\) and add to \(-8\). The numbers are \(-7\) and \(-1\).
4. Rewrite \(-8x\) as \(-7x - 1x\):
\[
7x^2 - 7x - 1x + 1
\]
5. Factor by grouping:
\[
7x(x - 1) - 1(x - 1)
\]
6. Combine the factors:
\[
(7x - 1)(x - 1)
\]
### (xi) \(2x^2 - 17x + 26\)
1. Identify \(A = 2\), \(B = -17\), and \(C = 26\).
2. Calculate \(A \cdot C = 2 \cdot 26 = 52\).
3. Find two numbers that multiply to \(52\) and add to \(-17\). The numbers are \(-13\) and \(-4\).
4. Rewrite \(-17x\) as \(-13x - 4x\):
\[
2x^2 - 13x - 4x + 26
\]
5. Factor by grouping:
\[
x(2x - 13) - 2(2x - 13)
\]
6. Combine the factors:
\[
(2x - 13)(x - 2)
\]
### (xii) \(3a^2 - 7a - 6\)
1. Identify \(A = 3\), \(B = -7\), and \(C = -6\).
2. Calculate \(A \cdot C = 3 \cdot (-6) = -18\).
3. Find two numbers that multiply to \(-18\) and add to \(-7\). The numbers are \(-9\) and \(2\).
4. Rewrite \(-7a\) as \(-9a + 2a\):
\[
3a^2 - 9a + 2a - 6
\]
5. Factor by grouping:
\[
3a(a - 3) + 2(a - 3)
\]
6. Combine the factors:
\[
(3a + 2)(a - 3)
\]
### (xiii) \(14a^2 + a - 3\)
1. Identify \(A = 14\), \(B = 1\), and \(C = -3\).
2. Calculate \(A \cdot C = 14 \cdot (-3) = -42\).
3. Find two numbers that multiply to \(-42\) and add to \(1\). The numbers are \(7\) and \(-6\).
4. Rewrite \(a\) as \(7a - 6a\):
\[
14a^2 + 7a - 6a - 3
\]
5. Factor by grouping:
\[
7a(2a + 1) - 3(2a + 1)
\]
6. Combine the factors:
\[
(2a + 1)(7a - 3)
\]
