Expanding binomial expressions is an important mathematical skill used in graphing parabolic shapes like the Sydney Harbour Bridge. There are two methods for expanding binomial expressions: using the order of operations (BODMAS/PEMDAS) or using the binomial expansion/FOIL method. Examples show how to apply the distribution property to expand binomial expressions with two, three, or four terms in the result. Expanding binomials is a fundamental skill needed for more advanced mathematics.

1. Image Source: http://www.strangezoo.com

2. Expanding Two Brackets is a skill needed for Graphing and Analysing
“Parabola” shapes, such as the Sydney Harbour Bridge.
Eg. It is a very basic and very important skill used in doing other parts
of mathematics, like dribbling is a fundamental skill of Basketball.
y = -0.00188(x – 251.5)(x – 251.5) + 118

3. The previous two brackets equation was created from measurements
taken off a photo of the Bridge, where pixels were converted to meters.

4. Expanding and simplifying the Two Brackets Equation, gives us an
Algebra equation we can graph using an online graphing application.

5. For the numeric expression below, there
are two ways we can get to the answer:
(2 + 3) (4 + 5) = 45
1) Using “BODMAS” or “Pemdas”
2) Using the “Binomial Expansion
or “FOIL” or “Crab Claws” method.

6. When we apply “BODMAS” or “Pemdas
to the expression below, we need to do the
Adding in the “Brackets” (or “Parenthesis”),
before we do the “Multiplying”.
(2 + 3) (4 + 5) = (2 + 3) x (4 + 5)
= 5 x 9
= 45

7. We use a set MULTIPLYING PATTERN
(2 + 3) (4 + 5) =
2 x 4 + 2 x 5 + 3 x 4 + 3 x 5
= 8 + 10 + 12 + 15
= 45Image Source: http://www.ceramicmosaicart.com

8. Image Source: http://www.ceramicmosaicart.com
To help remember the Pattern, think of the items in the first
bracket as two Crab Claws, which each reach into the second
bracket and grab the values there and multiply them.

9. 2(n + 5) = 2xn + 2x5 = 2n + 10
We cannot do Algebra expressions with
BODMAS, because n+3 does not
simplify to a whole number.
So we have to use Distributive Rule.
The “Crab Claws” is simply two lots of the
Distributive Rule one after each other.

10. Expand the Binomial: (h + 3) (y + 5)
(h + 3) (y + 5) =
h x y + h x 5 + 3 x y + 3 x 5
= hy + 5h + 3y + 15
Image Source: http://www.ceramicmosaicart.com

11. Expand the Binomial: (k + 2) (v - 1)
(k + 2) (v - 1) =
k x v + k x -1 + 2 x v + 2 x -1
= kv + -1k + 2v + -2
= kv - k + 2v - 2
Use Integer Rules
to Simplify each
+ - to just be a -

12. Expand the Binomial: (m - 2) (n - 6)
(m - 2) (n - 6) =
m x n + m x -6 + -2 x n + -2 x -6
= mn + -6m + -2n + 12
= mn - 6m - 2n + 12
Use Integer Rules
to Simplify each
+ - to just be a -

13. Expand the Binomial: (5a + h) (y + 2k)
(5a + h) (y + 2k) =
5a x y + 5a x 2k + h x y + h x 2k
= 5ay + 10ak + hy + 2hk

14. The examples we have done so far all had FOUR PART ANSWERS.
This is because the brackets had four different items in them:
(a + b) (c + d) = ac + ad + bc + bd
When we have brackets with some “Like Terms” in them,
we only get THREE PART or TWO PART answers.
(a + 3) (a + 2) = a2
+ 3a + 2a + 6 = a2
+ 5a + 6
(a + 4) (a - 4) = a2
+ -4a + 4a - 16 = a2
- 16
The following Examples show Three part and Two part Answers.
vv

15. Expand the Binomial: (m + 4) (m + 1)
(m + 4) (m + 1) =
m x m + m x 1 + 4 x m + 4 x 1
= m2
+ m + 4m + 4
= m2
+ 5m + 4
Simplify by Combining
the Like Term items.

16. Expand the Binomial: (b - 3) (b - 2)
(b - 3) (b - 2) =
b x b + b x -2 + -3 x b + -3 x -2
= b2
– 2b – 3b + 6
= b2
- 5b + 6
Simplify by Combining
the Like Term items.

17. Expand the Binomial: (h - 7) (h + 8)
(h - 7) (h + 8) =
h x h + h x 8 + -7 x h + -7 x 8
= h2
+ 8h – 7h + -56
= h2
+ h - 56
Simplify by Combining
the Like Terms .

18. Expand the Binomial: (k + 2) (k - 2)
(k + 2) (k - 2) =
k x k + k x -2 + 2 x k + 2 x -2
= k2
– 2k + 2k + -4
= k2
- 4
Simplify by Combining
the Like Terms to ZERO.

19. Expand the Binomial: (3m + 2) (2m + 1)
(3m + 2) (2m + 1) =
3m x 2m + 3m x 1 + 2 x 2m + 2 x 1
= 6m2
+ 3m + 4m + 2
= 6m2
+ 7m + 2
Simplify by Combining
the Like Terms .

20. Expand the Binomial: (a - 2)2
(a - 2)2
= (a - 2) (a - 2) =
a x a + a x -2 + -2 x a + -2 x -2
= a2
– 2a – 2a + 4
= a2
– 4a + 4
Simplify by Combining
the Like Terms .

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