What you need to know

Things to remember:

  • We expand brackets the same way as before, but this time we are multiplying through by a polynomial.
  • When multiplying polynomials with the same base, we add the powers.

Before we start, we will remind ourselves of two things:

  • Multiplying a single bracket by a constant (number).
  • Index rule for multiplication.

 

Expand 3(2x-5)

3(2x-5)=3\times2x-3\times5=6x-15

Write the following expressions as a single power.

 

Hint: Remember, if we multiply two terms with the same base number/letter, we add the powers.

 

x^4\times x^3=x^{4+3}=x^7

y^9\times y^{-4}=y^{9+-4}=y^{9-4}=y^5

x^4\times y^4=x^4y^4

We need to be careful with the third example, we can’t add the powers because the bases are different (x and y).

We can now put these two points together to expand brackets with a polynomial outside. Remember, polynomials are really just numbers that we don’t know yet, so we can multiply through as usual.

Expand 4x^3(2x^2-9y^5)

Step 1: Expand the bracket by writing as multiplications.

 4x^3(2x^2-9y^5)= 4x^3\times2x^2-4x^3\times 9y^5

 

Hint: Remember, the order of our multiplications don’t matter, so we can move the numbers next to each other.

Step 2: Rearrange to get the numbers together.

4x^3(2x^2-9y^5)= 4\times2\times x^3\times x^2-4\times9\times x^3\times y^5

Step 3: Do the multiplications.

4x^3(2x^2-9y^5)= 8\times x^5-36\times x^3y^5

Hint: Remember, we can’t add the powers if the bases are different, and we don’t need to write \times between numbers and variables.

4x^3(2x^2-9y^5)= 8x^5-36x^3y^5

Example Questions

Question 1: Expand 8x^3(2x^3-x)

Answer

Step 1: Expand the bracket by writing as multiplications.

 

 8x^3(2x^3-x)<em>=</em> 8x^3\times2x^3-8x^3\times x

 

 

Hint: Remember, the order of our multiplications don’t matter, so we can move the numbers next to each other.

 

Step 2: Rearrange to get the numbers together.

 

8x^3(2x^3-x)<em>=</em> 8\times2\times x^3\times x^3-8\times x^3\times x

 

Step 3: Do the multiplications.

 

8x^3(2x^3-x)<em>=</em> 16\times x^6-8\timesx^4

 

Hint: Remember, if we don’t see a power then it is 1, and we don’t need to write \times between numbers and variables.

 

8x^3(2x^3-x)<em>=</em> 16x^6-8x^4

Question 2: Expand 2x^2(x^4+5y^2)

Answer

Step 1: Expand the bracket by writing as multiplications.

 

 2x^2(x^4+5y^2)<em>=</em> 2x^2\times x^4+2x^2\times 5y^2[/

 

 

Hint: Remember, the order of our multiplications don’t matter, so we can move the numbers next to each other.

 

Step 2: Rearrange to get the numbers together.

 

2x^2(x^4+5y^2)<em>=</em> 2\times x^2\times x^4+2 \times 5\times x^2\times y^2

 

Step 3: Do the multiplications.

 

2x^2(x^4+5y^2)<em>=</em> 2\times x^6+ 10\times x^2\times y^2

 

Hint: Remember, we can’t add the powers if the bases are different, and we don’t need to write \times between numbers and variables.

 

2x^2(x^4+5y^2)<em>=</em> 2x^6+ 10x^2y^2