1. Grade 10 Mathematics
Products and rules

2. Table of Contents Multiply a binomial by a trinomial
Highest common factor
Difference of two squares
Factorise trinomials
Factorise by grouping
Simplify algebraic fractions with monomial denominators

3. Multiply a binomial by a trinomial
Theory of Products:
These rules form the basis of factorisation, which is the reverse of multiplication.
Multiply a binomial by a trinomial:
Always have the binomial as the first factor and then multiply all three terms in the second factor with the first term of the binomial. Then the second term with all three terms of the trinomial and then simplify. a +c
a b

4. Examples

5. Test your knowledge
Question 1 Simplify ( 3x +1)(9x2 – 3x +1) Answer A) 27x3 + 2x2 - 1 B) 9x3 – 2x2 +3x +1 C) 27x3 +1 D) None

6. Test your knowledge
Question 2 Simplify : (5x-1) (x2 +2x – 8) Answer A) 5x3 +9x2 - 35x + 8 B) 6x3 +32x – 8 C) 5x3 +5x2 - 35x + 8 D) 5x3 +9x2 - 42x + 8

7. Factorisation Steps for factorisation Look for
a) Highest Common Factor (HCF)
b) Difference of two Squares
c) Trinomial
d) Sum and Difference of two cubes
e) If there is brackets multiply out and rearrange from (1 – 4)

8. Highest common factor Examples Factorise 1) p2q2 – pq2 =pq2 (p – 1)
3a(p – 3q) – 2b(p – 3q)
=(p – 3q)(3a – 2b)
2p(a – b) – (a – b)
=(a – b)(2p -1)

9. Difference of two squares
The difference of two squares and perfect squares
1) a2-b2=(a-b)(a+b)
To identify the difference of two squares the following must be possible:
1. √T1
2. √ T2
3. There must be a minus sign between the two terms.

10. Examples 1) 2) 3) 4)

11. Factorise trinomials
An expression in which the highest exponent is 2 is a quadratic expression.
is called a trinomial because it contains three terms.
If we factorise this quadratic trinomial, we get two binomials as factors:
multiplication
Factorisation
Factorisation is the reverse of multiplication

12. Factorise trinomials
The sign of the last term indicates that the signs inside the two brackets will be the same e.g. the “+”sign in:
Write down the brackets with the correct signs. In this example both have to be negative.
Now check the factors of the constant term of which the sum add to 6, then choose the correct combination and factorise

13. Examples Factorise: One factor will be negative, one will be positive.
Combinations of factors:
The correct one must give – 8x
Therefore:
and this sum = 0

14. Test your knowledge
Question 3 Factorise the following: x2 + 14x +24 Answer A) (x -2) (x- 12) B) (x + 2) (x + 12) C) (x + 1) (x + 24) D) None of the above

15. Test your knowledge
Question 4 Factorise the following: x2 – 2x – 24 Answer A) (x – 4) (x + 6) B) (x – 12) (x + 2) C) (x + 4) (x + 6) D) (x + 4) (x - 6)

16. Solution B

17. Factorise by grouping
If we have more than three terms to factorise, we have to group either two-two or three – one together.
Always first check for a common factor in all four terms.
Sometimes we have to change the order by rearranging when grouping. Always change signs of each term when you place a bracket after a negative sign, because it is like taking out “– 1” as a common factor.
Remember 2 + c = c + 2

18. Test your knowledge Question 5 Factorise the following:
3xy – 3x + 3 – 3y
Answer
A) 3x(y – 1) B) 3(y + 1)(x + 1)
C) 3x(x + 1)(y – 1) D) 3(y – 1)(x – 1)

19. Test your knowledge
Question 6 Factorise the following: 4a2 – 2a + 9b2 – 3b + 12ab Answer A) (3a + 2b)(3a +2b +1) B) (2a +3b)( 2a – 3b + 1) C) (2a +3b) (2a + 3b – 1) D) None of the above

20. Simplify algebraic fractions with monomial denominators
We have to simplify fractions to its simplest form (equivalent fractions):
This means the numerator and denominator have no common factor.
Remember the rules for dividing with exponents.

21. Worked Examples: Remember: (x + 3y) = (3y + x)
Factorise by taking the common factor out and then simplify 3.
If we simplify fractions, we always factorise before cancelling

22. Test your knowledge Question 7 Simplify: 2y2 – 7xy + 3x2 2x2 - 4xy
Answer
A) 2y – x 2x
C) y+3x
-2x
B) (x - y)(y-3x) 2x
D) -(y-3x)