Polynomial Division (Edexcel A-Level Mathematics): Revision Notes

Example: $\frac{7}{2}$

• Quotient: $3$
• Remainder: $1$
• This representation tells us we can "pull out" three whole $2's$ from $7$, with a single unit being left over at the end. This remaining unit has not yet been divided by $2$ but needs to be.
• Therefore, $\frac{7}{2} = 3 + \frac{1}{2}$

Example: $\dfrac{x^2 + 3x + 5}{x + 1}$

• Step-by-step process:

1. Divide the leading term of the dividend $x^2$ by the leading term of the divisor $x$, which gives $x$.
2. Multiply the entire divisor by this result $x \cdot (x + 1) = x^2 + x$.
3. Subtract this result from the original polynomial: $(x^2 + 3x + 5) - (x^2 + x) = 2x + 5$.
4. Repeat the process with the new polynomial $(2x + 5)$.
5. Divide $2x$ by $x$ to get $2$.
6. Multiply the divisor by $2$: $2 \cdot (x + 1) = 2x + 2$.
7. Subtract this result from the new polynomial: $(2x + 5) - (2x + 2) = 3$.

• Result:
• Quotient: $x + 2$
• Remainder: $3$
• Therefore, $\dfrac{x^2 + 3x + 5}{x + 1} = x + 2 + \dfrac{3}{x + 1}$

Example: $\dfrac{x^3 + 5x^2 + 3x-4}{x + 5}$

Step-by-Step Solution:

1. Setup Polynomial Long Division:

• Dividend: $x^3 + 5x^2 + 3x - 4$
• Divisor: $x + 5$

2. First Division:

• Divide the leading term of the dividend $(x^3 )$ by the leading term of the divisor $( x )$, which gives $x^2$.
• Multiply $x^2$ by $x + 5 : x^2 \cdot (x + 5) = x^3 + 5x^2$.
• Subtract $x^3 + 5x^2$ from $x^3 + 5x^2 + 3x - 4$:
• Result: $(x^3 + 5x^2 + 3x - 4) - (x^3 + 5x^2) = 3x - 4$.

3. Second Division:

• Divide $3x$ by $x$ to get $3$.
• Multiply $3$ by $x + 5 : 3 \cdot (x + 5) = 3x + 15$ .
• Subtract $3x + 15$ from $3x - 4$ :
• Result: $(3x - 4) - (3x + 15) = -19$.

4. Result:

• Quotient: $x^2 + 3$
• Remainder: $-19$
• Therefore, the answer is:
• $\dfrac{x^3 + 5x^2 + 3x - 4}{x + 5} = x^2 + 3 - \dfrac{19}{x + 5}$

Final Answer:

$\boxed{x^2 + 3 - \frac{19}{x + 5}}$

The polynomial $f(x)$ is defined by $f(x) = x^3 - 9x^2 + 7x + 33$.

Example: Simplify $\dfrac{x^4 + 3x^3 + 9x - 12}{x+2}$

5. Set up the division:

6. Start dividing:

• Divide the first term of the numerator by the first term of the denominator: $\frac{x^4}{x} = x^3$.
• Multiply $x^3 by (x + 2)$ and subtract from the numerator:

• Divide $x^3$ by $x: \frac{x^3}{x} = x^2$.
• Multiply $x^2$ by $(x + 2)$ and subtract:

• Continue this process:

• The quotient ($Q$) and remainder ($r$) are:

7. Write the final answer:

Division Example: $\dfrac {2x^4-29x^2-40x-40}{x^2-3x-8}$

Must add Up $-40x$ and $-15x$

to $-40x \therefore$ $r=23x$

Need $-40$ in answer $\therefore$ no constant reminder