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Polynomial Division (CIE A Level Maths: Pure 3)
Revision Note
Polynomial Division
What is a polynomial?
- A polynomial is an algebraic expression consisting of a finite number of terms, with non-negative integer indices only
What is polynomial division?
- Polynomial division is a method for splitting polynomials into factor pairs (with or without an accompanying remainder term)
- At A level you will most frequently use it to factorise polynomials, or when dealing with improper (ie 'top-heavy') algebraic fractions
How do I divide polynomials?
- The method used for polynomial division is just like the long division method (sometimes called 'bus stop division') used to divide regular numbers:
- At A level you will normally be dividing a polynomial dividend of degree 3 or 4 by a divisor in the form (x ± p)
- The answer to a polynomial division question is built up term by term, working downwards in powers of the variable (usually x)
- Start by dividing by the highest power term
- Write out this multiplied by the divisor and subtract
- Continue to divide by each reducing power term and subtracting your answer each time
- Continue until you are left with zero
- If the divisor is not a factor of the polynomial then there will be a remainder term left at the end of the division
Worked example
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Quadratic Divisor
What are the degrees of the quotient and remainder when a polynomial is divided by a quadratic divisor?
- Suppose a polynomial of degree is divided by a quadratic divisor
- The quotient q will have degree
- The degree of the remainder r will be less than
- It could be degree 1 (linear)
- Or it could be degree 0 (constant)
How do I divide a polynomial by a quadratic divisor?
- You use polynomial division!
- Step 1
Divide the leading term of the polynomial by the squared term of the divisor- This gives the leading term of the quotient
- Step 2
Multiply this term by the divisor - Step 3
Subtract this from the polynomial to get a new polynomial with a lower degree - Continue these steps until you have an expression with a degree lower than 2
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