Chapter 4: Algebraic Expressions

Exercise 4.1 Solved Notes

Detailed, step-by-step solved reference guide for factorizing algebraic expressions, including common factors, pictorial factorization, and middle term splitting, completely cleaned of all watermarks.

Question 1 Common Factors
Factorize the following expressions by identifying common factors.

(i) 6x + 12

Solution
Write the expression: 6x + 12
Identify the greatest common divisor of coefficients 6 and 12, which is 6.
Factor out 6: 6(x + 2)
Answer: 6(x + 2)

(ii) 15y2 + 20y

Solution
Write the expression: 15y2 + 20y
Common numerical factor is 5. Common variable factor is y.
Factor out 5y: 5y(3y + 4)
Answer: 5y(3y + 4)

(iii) -12x2 - 3x

Solution
Write the expression: -12x2 - 3x
Identify the common factor as -3x.
Factor out -3x: -3x(4x + 1)
Answer: -3x(4x + 1)

(iv) 4a2b + 8ab2

Solution
Write the expression: 4a2b + 8ab2
Common factors are 4, a, and b, making the common monomial 4ab.
Factor out 4ab: 4ab(a + 2b)
Answer: 4ab(a + 2b)

(v) x2y - 3x2 + 2x

Solution
Write the expression: x2y - 3x2 + 2x
Identify the common factor for all three terms as x.
Factor out x: x(xy - 3x + 2)
Answer: x(xy - 3x + 2)

(vi) 3a2b - 9ab2 + 15ab

Solution
Write the expression: 3a2b - 9ab2 + 15ab
Identify the common factors: numerical 3, variables ab. Monomial common factor: 3ab.
Factor out 3ab: 3ab(a - 3b + 5)
Answer: 3ab(a - 3b + 5)
Question 2 Pictorial Representation
Factorize and represent the following expressions pictorially.

(i) 5x + 15

Solution
Factorize: 5(x + 3).
This represents a rectangle with a height of 5 units and a width of (x + 3) units.

Visual Diagram:

5x 15 x 3 5

(ii) x2 + 4x + 3

Solution
Factorize by splitting the middle term: x2 + 3x + x + 3 = x(x + 3) + 1(x + 3) = (x + 3)(x + 1).
This represents a rectangle with a width of (x + 3) and a height of (x + 1).

Area Decomposition Diagram:

x² 3x x 3 width = x + 3 height = x + 1

(iii) x2 + 6x + 8

Solution
Split the middle term: x2 + 4x + 2x + 8 = x(x + 4) + 2(x + 4) = (x + 4)(x + 2).
This represents a rectangle with a width of (x + 4) and a height of (x + 2).

Area Decomposition Diagram:

x² 4x 2x 8 width = x + 4 height = x + 2

(iv) x2 + 4x + 4

Solution
Factorize: x2 + 2x + 2x + 4 = x(x + 2) + 2(x + 2) = (x + 2)(x + 2) = (x + 2)2.
This represents a square with sides of length (x + 2).

Area Decomposition Diagram:

x² 2x 2x 4 width = x + 2 height = x + 2
Question 3 Trinomial Factorization
Factorize the following quadratic trinomials by splitting the middle term.

(i) x2 + x - 12

Solution
Identify constants: product = -12, sum = 1.
Find two numbers that multiply to -12 and add to 1: 4 and -3.
Split the middle term: x2 + 4x - 3x - 12
Group and factor: x(x + 4) - 3(x + 4) = (x + 4)(x - 3)
Answer: (x + 4)(x - 3)

(ii) x2 + 7x + 10

Solution
Multiply to 10, add to 7: the numbers are 5 and 2.
Split middle term: x2 + 5x + 2x + 10
Factor: x(x + 5) + 2(x + 5) = (x + 5)(x + 2)
Answer: (x + 5)(x + 2)

(iii) x2 - 6x + 8

Solution
Multiply to 8, add to -6: the numbers are -4 and -2.
Split: x2 - 4x - 2x + 8
Factor: x(x - 4) - 2(x - 4) = (x - 4)(x - 2)
Answer: (x - 4)(x - 2)

(iv) x2 - x - 56

Solution
Multiply to -56, add to -1: the numbers are -8 and 7.
Split: x2 - 8x + 7x - 56
Factor: x(x - 8) + 7(x - 8) = (x - 8)(x + 7)
Answer: (x - 8)(x + 7)

(v) x2 - 10x - 24

Solution
Multiply to -24, add to -10: the numbers are -12 and 2.
Split: x2 - 12x + 2x - 24
Factor: x(x - 12) + 2(x - 12) = (x - 12)(x + 2)
Answer: (x - 12)(x + 2)

(vi) y2 + 4y - 12

Solution
Multiply to -12, add to 4: the numbers are 6 and -2.
Split: y2 + 6y - 2y - 12
Factor: y(y + 6) - 2(y + 6) = (y + 6)(y - 2)
Answer: (y + 6)(y - 2)

(vii) y2 + 13y + 36

Solution
Multiply to 36, add to 13: the numbers are 9 and 4.
Split: y2 + 9y + 4y + 36
Factor: y(y + 9) + 4(y + 9) = (y + 9)(y + 4)
Answer: (y + 9)(y + 4)

(viii) x2 - x - 2

Solution
Multiply to -2, add to -1: the numbers are -2 and 1.
Split: x2 - 2x + x - 2
Factor: x(x - 2) + 1(x - 2) = (x - 2)(x + 1)
Answer: (x - 2)(x + 1)
Question 4 Advanced Trinomials
Factorize the following quadratic trinomials of the form ax2 + bx + c.

(i) 2x2 + 7x + 3

Solution
Find two numbers that multiply to ac = 6 and add to b = 7: the numbers are 6 and 1.
Split the middle term: 2x2 + 6x + x + 3
Group and factor: 2x(x + 3) + 1(x + 3) = (x + 3)(2x + 1)
Answer: (x + 3)(2x + 1)

(ii) 2x2 + 11x + 15

Solution
Multiply to ac = 30 and add to b = 11: the numbers are 6 and 5.
Split: 2x2 + 6x + 5x + 15
Factor: 2x(x + 3) + 5(x + 3) = (x + 3)(2x + 5)
Answer: (x + 3)(2x + 5)

(iii) 4x2 + 13x + 3

Solution
Multiply to ac = 12 and add to b = 13: the numbers are 12 and 1.
Split: 4x2 + 12x + x + 3
Factor: 4x(x + 3) + 1(x + 3) = (x + 3)(4x + 1)
Answer: (x + 3)(4x + 1)

(iv) 3x2 + 5x + 2

Solution
Multiply to ac = 6 and add to b = 5: the numbers are 3 and 2.
Split: 3x2 + 3x + 2x + 2
Factor: 3x(x + 1) + 2(x + 1) = (x + 1)(3x + 2)
Answer: (x + 1)(3x + 2)

(v) 3y2 - 11y + 6

Solution
Multiply to ac = 18 and add to b = -11: the numbers are -9 and -2.
Split: 3y2 - 9y - 2y + 6
Factor: 3y(y - 3) - 2(y - 3) = (y - 3)(3y - 2)
Answer: (y - 3)(3y - 2)

(vi) 2y2 - 5y + 2

Solution
Multiply to ac = 4 and add to b = -5: the numbers are -4 and -1.
Split: 2y2 - 4y - y + 2
Factor: 2y(y - 2) - 1(y - 2) = (y - 2)(2y - 1)
Answer: (y - 2)(2y - 1)

(vii) 4z2 - 11z + 6

Solution
Multiply to ac = 24 and add to b = -11: the numbers are -8 and -3.
Split: 4z2 - 8z - 3z + 6
Factor: 4z(z - 2) - 3(z - 2) = (z - 2)(4z - 3)
Answer: (z - 2)(4z - 3)

(viii) 6 + 7x - 3x2

Solution
Write expression in standard order: -3x2 + 7x + 6.
Multiply to ac = -18 and add to b = 7: the numbers are 9 and -2.
Split the middle term: 6 - 2x + 9x - 3x2
Factor by grouping: 2(3 - x) + 3x(3 - x) = (3 - x)(2 + 3x) = (3 - x)(3x + 2).
Answer: (3 - x)(3x + 2)
Interactive Sandbox Quadratic Trinomial Factorizer
Input coefficients for ax2 + bx + c to split the middle term and factorize the expression step-by-step.
Step-by-step Solution