Long Division (Junior Cert Mathematics): Revision Notes

Example Problem 1:

Divide $x^3$ by $x$ and put the answer at the top, $x^2$

Multiply $x - 2$ by $x^2$ and put the answer under the first two terms

Subtract (change signs) and divide $5x^2$ by $x$

Multiply $x - 2$ by $5x$

Subtract (change signs) and divide $6x$ by $x$

Multiply $x - 2$ by $6$

Subtract (change signs)

Answer

$(x^2 + 5x + 6)$

We are basically repeating the same step 3 times. You will know that your answer is correct if when you subtract the last set of terms your answer is 0.

Example Problem 2: If we are asked to divide into an expression that has some parts missing, for example there is no $x^2$ part, we leave space for any that may appear.

Divide $27x^3$ by $3x$ and put the answer at the top, $9x^2$

Multiply $3x - 1$ by $9x^2$ and put answer underneath

Subtract (change signs) and divide $9x^2$ by $3x$

Multiply $3x - 1$ by $3x$

Subtract (change signs) and divide $3x$ by $3x$

Multiply $3x - 1$ by $1$

Subtract (change signs)

Answer

$(9x^2 + 3x + 1)$

Example Problem 3: Simplify$\frac{x^3 - 5x^2 + 10x - 12}{x - 3} .$

Step 1: Set Up the Division

Write the division as you would in a long division problem with numbers: $\text{Divide } x^3 - 5x^2 + 10x - 12 \text{ by } x - 3$

Step 2: Divide

First Division:

• Divide the first term of the dividend $x^3$ by the first term of the divisor $(x)$: $\frac{x^3}{x} = x^2$
• Write $x^2$ above the division line.

Step 3: Multiply

• Multiply $x^2$ by the entire divisor $x - 3$: $x^2 \times (x - 3) = x^3 - 3x^2$
• Write this product under the first two terms of the dividend: $x^3 - 5x^2 \quad \text{(write)} \quad x^3 - 3x^2$

Step 4: Subtract

• Subtract the result from the previous step from the dividend: $(x^3 - 5x^2) - (x^3 - 3x^2) = -2x^2$

Step 5: Bring Down

• Bring down the next term from the original dividend which is $10x$: $-2x^2 + 10x$

Step 6: Repeat the Process

Second Division:

• Now, divide the first term of the new expression $-2x^2$ by the first term of the divisor $(x)$: $\frac{-2x^2}{x} = -2x$

• Write $-2x$ above the division line next to $x^2$. Second Multiplication:

• Multiply $-2x$ by the entire divisor $x - 3$: $-2x \times (x - 3) = -2x^2 + 6x$

• Write this product under the current terms: $-2x^2 + 10x \quad \text{(write)} \quad -2x^2 + 6x$ Second Subtraction:

• Subtract the product from the expression above: $(-2x^2 + 10x) - (-2x^2 + 6x) = 4x$ Second Bring Down:

• Bring down the next term from the original dividend which is $-12$: $4x - 12$

Step 7: Final Division

Third Division:

• Divide the first term of the new expression $4x$ by the first term of the divisor $x$: $\frac{4x}{x} = 4$

• Write $4$ above the division line next to $-2x$. Third Multiplication:

• Multiply $4$ by the entire divisor $(x - 3)$: $4 \times (x - 3) = 4x - 12$

• Write this product under the current terms: $[ 4x - 12 \quad \text{(write)} \quad 4x - 12 ]$ Third Subtraction:

• Subtract the product from the expression above: $(4x - 12) - (4x - 12) = 0$

Step 8: Final Answer

• Since there's no remainder, the final answer is: $x^2 - 2x + 4$

Example Problem 4:

Exam Tip: Keep your work organised! Writing each step clearly will help you avoid mistakes and make it easier to spot any errors. Write like terms beneath like terms i.e. the x beneath the x when brought underneath etc.