Expanding Brackets Part 1

Video 1

Video 2

Expanding expressions with 1 bracket

Expanding brackets is multiplying the outside terms with EACH term inside the brackets.
Lets try this question.
Expand a(a + b)
So first you would multiply a with a, then you would multiply
a x a = a²a x b = ab
So a(a + b) = (a x a) + (a x b) = a² + ab
Lets try another question which is seen below.
Expand a(a)
a x a = a²
So a(a) = a²

Video 1 Explanation

Question 1: Expand 5(a - 7)
5(a - 7)
So the 5 (which is the outside the bracket) needs to be multiplied with the terms inside the bracket one by one.
5 x a = 5a5 x -7 = -35 (Remember a positive number multiplied by a negative number is a negative number)
So
5(a - 7) = (5 x a) + (5 x -7) = 5a + -35 = 5a - 35
Question 2: Expand a(2 + b)
So the a (which is the outside the bracket) needs to be multiplied with the terms inside the bracket one by one.
a x 2 = 2aa x b = ab
So
a(2 + b) = (a x 2) + (a x b) = 2a + ab
Lets try another question which is seen below.
Expand (a - cd)d
Remember (a + cd)d = (a + cd) x d = d x (a + cd)Remember when multiplying, we can switch what comes first when writing out the product e.g.
5 x 4 = 4 x 5 (ITS THE SAME)
So now we can expand.
(a - cd)d = d(a - cd) = da - cd²
Why?
d x a = da
d x cd = c x d x d = cd²

Video 2 Explanation (Double Brackets)

Expanding double brackets is just double the work, if you can expand 1 bracket you can definitely expand 2 brackets.
Step by step like earlier, multiply the first term in the first bracket, with each term in the second bracket. Then multiply the second term in the first bracket with each term in the second bracket. MAKE SURE TO BE CAREFUL AND TAKE INTO CONSIDERATION WHETHER THE TERMS ARE POSITIVE OR NEGATIVE WHEN MULTIPLYING. After you attain all the products, add them together. This may sound hard to read but once you see below the calculations you will understand.
Question 1: Expand (x + y)(a + b)
x X a = xax X b = xb
y X a = ya y X b = yb
(We have used X to symbolise the multiplication sign, so you don't confuse it with x which is the term in the bracket)
So
(x + y)(a + b) = xa + xb + ya + yb
When we expand the letter x with the terms in the second bracket, a and b, we have actually expanded x(a + b).
Look at the calculations:
x(a + b) = xa + xb
When we expand the letter y with the terms in the second bracket, a and b, we have actually expanded y(a + b).
Look at the calculation:
y(a + b) = ya + yb
so
(x + y)(a + b) = x(a + b) + y(a + b) = xa + xb + ya + yb
So that is why we said expanding double brackets is just double the work.
Question 2: Expand (2 + a)(a - 3)
2 x a = 2a2 x -3 = -6a x a = a^2a x -3 = -3a
so
(2 + a)(a - 3) = 2(a - 3) + a(a -3) = 2a - 6 + a² - 3a
BUT WAIT
We can simplify 2a - 6 + a² - 3a by collecting like terms
so
2a - 6 + a² - 3a = -a - 6 + a²
Lets try another question which is seen below.
Expand (-ab + c + c)(3 + b²)
(-ab + c + c)(3 + b²) = (-ab + c + c) x (3 + b²)
We can see that we can simplify the first bracket as (-ab + c + c) = (-ab + 2c)
So
(-ab + c + c)(3 + b^2) = (-ab + 2c)(3 + b²)
-ab x 3 = -3ab-ab x b² = -a x b x b x b = -ab³ (Remember -ab = -a x b)2c x 3 = 6c2c x b² = 2 x c x b x b = 2cb²
so
(-ab + 2c)(3 + b²) = -3ab -ab³ + 6c + 2cb²

Try these questions

Expand 4(a + 6 - y²)
Expand 3ay(15 + y^-1)
Expand (a + 3)(a + 2) (REMEMBER TO SIMPLIFY)
Expand (2b + 4)(b + 3) (REMEMBER TO SIMPLIFY)

Answers

4(a + 6 - y²) = 4a + 24 -4y²

4 x a = 4a 4 x 6 = 244 x -y² = -4y²

3ay(15 + y^-1) = 45ay + 3a

3ay x 15 = 45ay3ay x y^-1= 3 x a x y x y^-1 = 3 x a x 1 = 3a
(We can add the powers when multiplying, y = y¹, so y¹ x y^-1 = y^{(1 + - 1)}= y⁰= 1 (Anything to the power of 0 is 1)
If you do not understand why we added the powers, check the indices part 2 page)
so 3ay(15 + y^-1) = 45ay + 3a

(a + 3)(a + 2) = a² + 5a + 6

a x a = a²a x 2 = 2a3 x a = 3a3 x 2 = 6
a² + 2a + 3a + 6 = a² + 5a + 6

(2b + 4)(b + 3) = 2b² + 10b + 12

2b x b = 2 x b x b = 2b²2b x 3 = 6b4 x b = 4b4 x 3 = 12
2b² + 6b + 4b + 12 = 2b² + 10b + 12

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