Notes · Unit 4
Factorization and Algebraic Manipulation
Thirteen short slides on factorization, identities, HCF and LCM, square roots and algebraic fractions.
Press Next (or the right arrow key) to move on. On most slides, each press reveals one more idea.
IdentitiesTrinomialsCubesHCF and LCMAlgebraic fractions
1 · Factorization
Write an expression as a product
Factorization is the opposite of multiplying out. For example, 5x and 3xy are factors of 15x2y because their product gives it.
Difference of two squares
a2 − b2 = (a + b)(a − b)
So x2 − 4 = (x + 2)(x − 2). Note that x − 4 is not a factor.
a2 − b2 = (a + b)(a − b)
So x2 − 4 = (x + 2)(x − 2). Note that x − 4 is not a factor.
Perfect square trinomials
a2 + 2ab + b2 = (a + b)2 and a2 − 2ab + b2 = (a − b)2
a2 + 2ab + b2 = (a + b)2 and a2 − 2ab + b2 = (a − b)2
Check the middle term. In 4x2 + 12x + 9, the middle term is 2 × (2x) × 3 = 12x, so it equals (2x + 3)2.
2 · Trinomials
Split the middle term
For x2 + px + q, find two numbers whose product is q and whose sum is p.
x2 + 5x + 6: the numbers are 2 and 3.
= (x + 2)(x + 3)
= (x + 2)(x + 3)
x2 − 7x + 12: the product 12 is positive, so both numbers have the same sign. The sum is negative, so both are negative: −3 and −4.
= (x − 3)(x − 4)
= (x − 3)(x − 4)
2x2 − 5x − 3: split −5x as −6x + x.
= 2x(x − 3) + 1(x − 3) = (2x + 1)(x − 3)
= 2x(x − 3) + 1(x − 3) = (2x + 1)(x − 3)
3x2 + 11x + 6: split 11x as 9x + 2x.
= 3x(x + 3) + 2(x + 3) = (3x + 2)(x + 3)
= 3x(x + 3) + 2(x + 3) = (3x + 2)(x + 3)
3 · Cubes
Sum and difference of cubes
Sum of cubes
a3 + b3 = (a + b)(a2 − ab + b2)
a3 + b3 = (a + b)(a2 − ab + b2)
Difference of cubes
a3 − b3 = (a − b)(a2 + ab + b2)
a3 − b3 = (a − b)(a2 + ab + b2)
Cube of a sum
(a + b)3 = a3 + 3a2b + 3ab2 + b3
(a + b)3 = a3 + 3a2b + 3ab2 + b3
Example. x3 + 27 = x3 + 33 = (x + 3)(x2 − 3x + 9)
Careful. (x − y)3 ≠ x3 − y3. The left side expands to x3 − 3x2y + 3xy2 − y3.
4 · Clever methods
Turn it into something you know
Group into a square. a2 + 2ab + b2 − c2 = (a + b)2 − c2 = (a + b + c)(a + b − c)
Add and subtract a term. a4 + 4b4 = (a2 + 2b2)2 − (2ab)2 = (a2 + 2ab + 2b2)(a2 − 2ab + 2b2)
Add and subtract a2b2. a4 + a2b2 + b4 = (a2 + b2)2 − (ab)2 = (a2 + ab + b2)(a2 − ab + b2)
Substitute. For (x2 + 3x − 4)(x2 + 3x + 5) + 8, put a = x2 + 3x.
(a − 4)(a + 5) + 8 = a2 + a − 12 = (a + 4)(a − 3).
Put back a: (x2 + 3x + 4)(x2 + 3x − 3).
(a − 4)(a + 5) + 8 = a2 + a − 12 = (a + 4)(a − 3).
Put back a: (x2 + 3x + 4)(x2 + 3x − 3).
5 · HCF and LCM
Factorize first, then compare
HCF
Take the common factors, each with its lowest power.
LCM
Take every factor, each with its highest power.
Monomials. 12x2y and 18xy2: HCF of 12 and 18 is 6, lowest powers x and y, so the HCF is 6xy.
6x2y and 9xy3: LCM of 6 and 9 is 18, highest powers x2 and y3, so the LCM is 18x2y3.
6x2y and 9xy3: LCM of 6 and 9 is 18, highest powers x2 and y3, so the LCM is 18x2y3.
Polynomials. x2 − x − 6 = (x − 3)(x + 2) and x2 + x − 2 = (x + 2)(x − 1).
LCM = (x − 3)(x + 2)(x − 1).
LCM = (x − 3)(x + 2)(x − 1).
Powers matter. a2 − b2 = (a + b)(a − b) and (a − b)2 have HCF a − b.
6 · Useful facts
HCF and LCM together
HCF × LCM = P × Q
If the HCF of P and Q is 1, then LCM = P × Q.
Multiplying one polynomial by a non-zero constant does not change the HCF. If the HCF of P and Q is x2 + 2x, the HCF of 5P and Q is the same.
Division method for HCF. You may multiply by a non-zero constant to avoid fractions, because that adds no new polynomial factor.
Do not multiply by a variable. It can bring in a new common factor and give a wrong HCF.
7 · Square roots
Square root of an expression
If P = Q × Q, then Q is a square root of P.
Both +Q and −Q work, so the square roots are ±Q.
Example. 4a2 + 12ab + 9b2 = (2a + 3b)2, so the square roots are ±(2a + 3b).
Method. Factorize into a perfect square, then take one factor.
8 · Algebraic fractions
Simplify, multiply and divide
The denominator can never be zero. For P/Q, we need Q ≠ 0.
Factorize, then cancel. (x2 − 9)/(x − 3) = (x + 3)(x − 3)/(x − 3) = x + 3, for x ≠ 3.
Dividing means multiplying by the reciprocal.
(x2 − 1)/(x + 2) ÷ (x − 1)/(x2 − 4) = (x + 1)(x − 1)/(x + 2) × (x + 2)(x − 2)/(x − 1) = (x + 1)(x − 2)
(x2 − 1)/(x + 2) ÷ (x − 1)/(x2 − 4) = (x + 1)(x − 1)/(x + 2) × (x + 2)(x − 2)/(x − 1) = (x + 1)(x − 2)
For products, factorize all numerators and denominators first, so you can cancel to the most reduced form.
9 · Adding and subtracting
Use the LCM as the common denominator
With unlike denominators, first find the LCM of the denominators and rewrite each fraction.
3/(x − 1) + 2/(x + 1)
The LCM is x2 − 1.
= (3(x + 1) + 2(x − 1))/(x2 − 1) = (5x + 1)/(x2 − 1)
The LCM is x2 − 1.
= (3(x + 1) + 2(x − 1))/(x2 − 1) = (5x + 1)/(x2 − 1)
Order of operations. DMAS: division, multiplication, addition, subtraction.
Complex fraction. A fraction whose numerator, denominator or both contain algebraic expressions.
10 · Common mistakes
Five traps to avoid
Cancelling terms. (x + 2)/x ≠ 2. Only common factors cancel. Here the fraction equals 1 + 2/x.
Missing middle term. x2 + 9 ≠ (x + 3)2, because (x + 3)2 = x2 + 6x + 9.
Forgetting the restriction. After simplifying (x2 − 9)/(x − 3) to x + 3, remember x ≠ 3, since the original fraction is undefined there.
Wrong sign in a cube factor. The bracket for a3 + b3 has a2 − ab + b2, the opposite sign to the one in the first factor.
Multiplying by a variable in the division method for HCF. Use only a non-zero constant.
11 · Quick check
Try these, then tap to check
12 · Your turn
Practise Unit 4
Three timed MCQ tests with an explanation after every answer.
Practice questions are independently written and are not official FBISE papers.