1. Starter Questions x 2a 3b 4ab 4 7 a b 3a 5b 8a2b 21ab2
Complete the following multiplication grid:
Red:
Amber:
Green:
Extension: x 2a 3b
4ab 4 7 a b 3a 5b
8a2b
21ab2
Hint: for the extension question, try solving
___ x 2a = 8a2b and ___ x 3b = 21ab2 to fill the first column.

2. Answers x 2a 3b 4ab 4 8a 12b 16ab 7 14a 21b 28ab a 2a2 3ab 4a2b b 2ab

3. Everything outside the brackets must multiply everything inside the brackets.
Expand 4 (3y + 7)
Expand 3x (2y - 1)
= 12y
+ 28
= 6xy
- 3x

4. Answers 2a – 6 3b + 15 2c – 2 3d + 3 10e – 20 8 – 12f g² + 4g h² - 3h
2k² + 5k 3mn – 2m
2pq + 12p 6r – 6rs

5. Expand and simplify:
a(b - a) + a(a + 2)
= ab
- a²
+ a²
+ 2a
= ab + 2a
m(m – 2n) + 3m(5n + 3m)
= m²
- 2mn
+ 15mn
+ 9m²
= 10m² + 13mn

6. Answers 24x – 18 + 12x – 3 = 36x – 21 25z + 60 + 15z – 6 = 40z + 54
5y² + 6y + y² + 5y = 6y² + 11y
5r² + pr + 2r² + 2pr = 7r² + 3pr
4st + 2ps + 2st + 2pt = 6st + 2ps + 2pt
12ab + 4ap + 2a² + 2ap = 12ab + 6ap + 2a²

7. Starter
Expand and simplify (where possible):
6 (a + b)
x (x + y)
4 (m + 2n) + 3 (m – n)
2x (x + 3y) – 3x (x – y)
= 6a + 6b
= x² + xy
= 7m + 5n
= -x² + 9xy

8. There are two different methods we can use to expand these brackets.
(p + 3)(p – 7)
There are two different methods we can use to expand these brackets.

9. Grid Method (p + 3)(p – 7) x p +3 -7 p² +3p -7p -21 p² - 7p + 3p - 21

10. FOIL Method (p + 3)(p – 7) First: p² Outside: -7p Inside: +3p
Last: -21
p² - 7p + 3p - 21
p² - 4p - 21

11. Questions 1. (x + 4)(x + 2) 2. (x + 7)(x + 8) 3. (x + 3)(x + 2)

12. Answers 1. x² + 6x + 8 2. x² + 15x + 56 3. x² + 5x + 6

13. What is different about this example?
(4y - 3)(2y – 5)
What is different about this example?

14. Grid Method (4y - 3)(2y – 5) x 4y -3 2y -5 8y² -6y -20y +15
8y² - 26p + 15

15. FOIL Method (4y - 3)(2y – 5) First: 8y² Outside: -20y Inside: -6y
Last: +15
8y² - 20y - 6y + 15
8y² - 26p + 15

16. Questions 1. (2x + 4)(x + 2) 2. (x + 7)(3x + 8) 3. (5x + 3)(x + 2)

17. Answers 1. 2x² + 8x + 8 2. 3x² + 29x + 56 3. 5x² + 13x + 6

18. Sketching Graphs (x + 3)(x – 7) = x² - 4x - 21
x² tells us the shape of the graph
-21 tells us the y-intercept y
-21

19. These are the x-intercepts
Sketching Graphs
(x + 3)(x – 7) = 0
(x + 3) = 0
So x = -3
(x - 7) = 0
So x = 7
These are the x-intercepts

20. Sketch the graphs for the equations you have just expanded.
Sketching Graphs
y = (x + 3)(x – 7) y
y-intercept = -21
x-intercepts = -3, 7
Sketch the graphs for the equations you have just expanded. x -3 7
-21

21. Starter Where do these graphs cross the x-axis? Expand the brackets
Where do these graphs cross the y-axis?
y = (x – 4)(x + 2)
y = (x – 7)(x – 3)
y = (x + 5)²

22. Where do these graphs cross the x-axis?
Expand the brackets
Where do these graphs cross the y-axis?
1. y = (x – 4)(x + 2)
x = 4 or -2
y = x² - 2x - 8
y-intercept at -8

23. Where do these graphs cross the x-axis?
Expand the brackets
Where do these graphs cross the y-axis?
2. y = (x – 7)(x - 3)
x = 7 or 3
y = x² - 10x + 21
y-intercept at +21

24. Where do these graphs cross the x-axis?
Expand the brackets
Where do these graphs cross the y-axis?
3. y = (x + 5)²
x = -5
y = x² + 10x + 25
y-intercept at +25

25. If the greatest power of x is 1, the equation is linear.
Eg: y = 4x - 1
If the greatest power of x is 2 (x²), the equation is quadratic.
Eg: y = x² + 6x – 2
If the greatest power of x is 3 (x³), the equation is cubic.
Eg: y = x³ - 5x² + 4x - 9

26. True or False:
y = 3x² + 6x – 2 is a cubic equation
y = 2x³ - 9x is a cubic equation
y = 5 – 3x is a linear equation
y = 3 – 7x³ is a quadratic equation

27. Show me:
A linear equation where x = 2 and y = 7
A quadratic equation where x = 2 and y = 7
A cubic equation where x = 2 and y = 7

28. Remember the solutions of x are where the graph crosses the x-axis!
Where does the graph y = (x + 4)(x + 1)(x – 2) cross the x-axis and the y-axis? 
Remember the solutions of x are where the graph crosses the x-axis!
If (x + 4)(x + 1)(x – 2) = 0,
x = -4, -1 or 2
To find where the graph crosses the y-axis, we’ll need to expand the brackets…

29. (x + 4)(x + 1)(x – 2)
Start by expanding one pair of brackets. x +1 -2 x² +x
-2x -2
= x² - 2x + x - 2
= x² - x - 2
So (x + 4)(x + 1)(x – 2) = (x + 4)(x² - x – 2)

30. (x + 4)(x + 1)(x – 2) = (x + 4)(x² - x – 2)
Now we can expand the rest. x x² -x -2 +4 x³
-x²
-2x
+4x²
-4x -8
= x³ + 4x² - x² - 4x – 2x - 8
= x³ + 3x² - 6x - 8
So (x + 4)(x + 1)(x – 2) = x³ + 3x² - 6x - 8

31. Where does the graph y = (x + 4)(x + 1)(x – 2) cross the x-axis and the y-axis? 
If (x + 4)(x + 1)(x – 2) = 0,
x = -4, -1 or 2
(x + 4)(x + 1)(x – 2) = x³ + 3x² - 6x - 8
The y-intercept is -8.

32. Where does the graph y = (2x - 1)(x + 3)(x + 5) cross the x-axis and the y-axis? 
If (2x - 1)(x + 3)(x + 5) = 0,
x = ½, -3 or -5
Start by expanding one pair of brackets.
So (2x - 1)(x + 3)(x + 5) = (2x - 1)(x² + 8x + 15)
Now we can expand the rest.
So (2x - 1)(x + 3)(x + 5) = 2x³ + 15x² + 22x - 15
Show expansion of brackets on whiteboard to model working
The y-intercept is -15.

33. Where does the graph y = 5x(2x - 3)(x - 2) cross the x-axis and the y-axis? 
If 5x(2x - 3)(x - 2) = 0,
x = 0, 3/2 or 2
Start by expanding one pair of brackets.
So 5x(2x - 3)(x - 2) = 5x(2x² - 7x + 6)
Now we can expand the rest.
So 5x(2x - 3)(x - 2) = 10x³ - 35x² + 30x
Show expansion of brackets on whiteboard to model working
The y-intercept is 0.

34. Answers

35. Answers 3. Expand the following expressions.
a. (x + 2)(x + 5)(x + 1) b. (x – 3)(x + 4)(x – 2)
= x³ + 8x² + 17x = x³ - x² - 14x + 24
c. x(x + 5)(x – 4) d. (x + 3)³
= x³ + x² - 20x = x³ + 9x² + 27x + 27
4. What equations do these graphs represent?
y = (x + 3)(x + 1)(x – 1), y = (x + 1)(x – 1)(x – 2), y = (x + 1)(x – 2)(x – 3)
5. How many solutions does each cubic equation have?
a b. 1
c d. 2
e f. 3