## 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 $8x^3(2x^3-x)$

Step 1: Expand the bracket by writing as multiplications.

$$8x^3(2x^3-x)= 8x^3\times2x^3-8x^3\times x$$

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.

$$8x^3(2x^3-x)= 8\times2\times x^3\times x^3-8\times x^3\times x$$

Step 3: Do the multiplications.

$$8x^3(2x^3-x)= 16\times x^6-8\timesx^4$$

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.

$$8x^3(2x^3-x)= 16x^6-8x^4$$

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

Step 1: Expand the bracket by writing as multiplications.

$$2x^2(x^4+5y^2)= 2x^2\times x^4+2x^2\times 5y^2[/$$

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.

$$2x^2(x^4+5y^2)= 2\times x^2\times x^4+2 \times 5\times x^2\times y^2$$

Step 3: Do the multiplications.

$$2x^2(x^4+5y^2)= 2\times x^6+ 10\times x^2\times y^2$$

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.

$$2x^2(x^4+5y^2)= 2x^6+ 10x^2y^2$$