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Factor & Remainder Theorem (Cambridge (CIE) O Level Additional Maths): Revision Note

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Factor Theorem

What is the factor theorem?

  • The factor theorem is a useful result concerning the roots and factors of polynomials

    • In the example below, the polynomial 4 x cubed plus 8 x squared minus 9 x minus 18 has three (linear) factors

      • open parentheses x plus 2 close parentheses comma space open parentheses 2 x plus 3 close parentheses and open parentheses 2 x minus 3 close parentheses

      • and so it has the three roots x equals negative 2 comma space x equals negative 3 over 2 and x equals 3 over 2

Factorised polynomial with 3 factors

  • For a polynomial straight f open parentheses x close parentheses the factor theorem states that:

i) if straight f open parentheses p close parentheses equals 0, then open parentheses x minus p close parentheses is a factor of straight f open parentheses x close parentheses
(x equals p is a root of straight f open parentheses x close parentheses)

and

ii) if open parentheses x minus p close parentheses is a factor of straight f open parentheses x close parentheses, then straight f open parentheses p close parentheses equals 0

explanation of the factor theorem

Examiner Tips and Tricks

  • In an exam, the values of p you'll need to find that make straight f open parentheses p close parentheses equals 0 are going to be integers close to zero 

    • Try p equals 1 and p equals negative 1 first, then 2 and -2, then 3 and -3

    • It is unlikely that you'll have to go beyond that

Worked Example

a) Show that open parentheses x minus 2 close parentheses is a factor of the polynomial straight f open parentheses x close parentheses equals x cubed plus 6 x squared minus 9 x minus 14.

(From part (ii) of our definition of factor theorem ...) ... if open parentheses x minus 2 close parentheses is a factor of straight f open parentheses x close parentheses then straight f open parentheses 2 close parentheses equals 0.

table row cell straight f open parentheses 2 close parentheses end cell equals cell open parentheses 2 close parentheses cubed plus 6 open parentheses 2 close parentheses squared minus 9 open parentheses 2 close parentheses minus 14 end cell row cell straight f open parentheses 2 close parentheses end cell equals cell 8 plus 24 minus 18 minus 14 end cell row cell straight f open parentheses 2 close parentheses end cell equals 0 end table

Since bold f stretchy left parenthesis 2 stretchy right parenthesis bold equals bold 0, stretchy left parenthesis bold italic x minus 2 stretchy right parenthesis is factor of .

b) Use the factor theorem to find another factor of straight f open parentheses x close parentheses.

Try straight f open parentheses 1 close parentheses first,

table attributes columnalign right center left columnspacing 0px end attributes row cell straight f open parentheses 1 close parentheses end cell equals cell open parentheses 1 close parentheses cubed plus 6 open parentheses 1 close parentheses squared minus 9 open parentheses 1 close parentheses minus 14 end cell row cell straight f open parentheses 1 close parentheses end cell equals cell 1 plus 6 minus 9 minus 14 end cell row cell straight f open parentheses 1 close parentheses end cell equals cell negative 16 end cell end table

Since straight f open parentheses 1 close parentheses not equal to 0, open parentheses x minus 1 close parentheses is not a factor of straight f open parentheses x close parentheses.

Try straight f open parentheses negative 1 close parentheses,

table row cell straight f open parentheses negative 1 close parentheses end cell equals cell open parentheses negative 1 close parentheses cubed plus 6 open parentheses negative 1 close parentheses squared minus 9 open parentheses negative 1 close parentheses minus 14 end cell row cell straight f open parentheses negative 1 close parentheses end cell equals cell negative 1 plus 6 plus 9 minus 14 end cell row cell straight f open parentheses negative 1 close parentheses end cell equals 0 end table

Since straight f open parentheses negative 1 close parentheses equals 0, open parentheses x plus 1 close parentheses is a factor of straight f open parentheses x close parentheses.

stretchy left parenthesis bold italic x plus 1 stretchy right parenthesis is another factor of .

open parentheses x plus 7 close parentheses is the third (linear) factor.
Once one factor is known, polynomial division could be used to find the others. (In this case we were specifically asked to use factor theorem.)

Remainder Theorem

What is the remainder theorem?

  • The factor theorem is actually a special case of the more general remainder theorem

  • The remainder theorem states that when the polynomial straight f open parentheses x close parentheses is divided by open parentheses x minus a close parentheses the remainder is straight f open parentheses a close parentheses

    • You may see this written formally as straight f open parentheses x close parentheses equals open parentheses x minus a close parentheses straight Q open parentheses x close parentheses plus straight f open parentheses a close parentheses

    • In polynomial division

      • straight Q open parentheses x close parentheses would be the result (at the top) of the division (the quotient)

      • straight f open parentheses a close parentheses would be the remainder (at the bottom)

      • open parentheses x minus a close parentheses is called the divisor

    • In the case when straight f open parentheses a close parentheses equals 0 comma space straight f open parentheses x close parentheses equals open parentheses x minus a close parentheses straight Q open parentheses x close parentheses and hence open parentheses x minus a close parentheses is a factor of straight f open parentheses x close parentheses – the factor theorem!

How do I solve problems involving the remainder theorem?

  • If it is the remainder that is of particular interest, the remainder theorem saves the need to carry out polynomial division in full

    • e.g.  The remainder from left parenthesis x squared minus 2 x right parenthesis divided by left parenthesis x minus 3 right parenthesis is 3 squared minus 2 cross times 3 equals 3

    • This is because if straight f open parentheses x close parentheses equals x squared minus 2 x and a equals 3

  • If the remainder from a polynomial division is known, the remainder theorem can be used to find unknown coefficients in polynomials

    • g. The remainder from left parenthesis x squared plus p x right parenthesis divided by left parenthesis x minus 2 right parenthesis is 8 so the value of p can be found by solving 2 squared plus p open parentheses 2 close parentheses equals 8, leading to p space equals space 2

    • In harder problems there may be more than one unknown in which case simultaneous equations would need setting up and solving

  • The more general version of remainder theorem is if straight f open parentheses x close parentheses is divided by open parentheses a x minus b close parentheses then the remainder is  begin mathsize 16px style straight f stretchy left parenthesis b over a stretchy right parenthesis end style

    • The remainder is still found by evaluating the polynomial at the value of x such that a x minus b equals 0 (the divisor is zero) but it is not necessarily an integer

Examiner Tips and Tricks

  • Exam questions will use formal mathematical language which can make factor and remainder theorem questions sound more complicated than they are

    • Ensure you are familiar with the various terms from these revision notes

Worked Example

The polynomial straight p open parentheses x close parentheses is given by 8 x to the power of 4 plus a x squared plus b x minus 1, where a and b are integer constants.
When straight p open parentheses x close parentheses is divided by open parentheses x minus 1 close parentheses the remainder is 9.
When straight p open parentheses x close parentheses is divided by open parentheses 2 x minus 1 close parentheses the remainder is 1.
Find the values of a and b.

Remainder theorem: "straight f open parentheses a close parentheses is the remainder when straight f open parentheses x close parentheses is divided by open parentheses x minus a close parentheses".

x minus 1 equals 0 when x equals 1:

table attributes columnalign right center left columnspacing 0px end attributes row cell straight p open parentheses 1 close parentheses end cell equals 9 row cell 8 plus a plus b minus 1 end cell equals 9 row cell a plus b end cell equals 2 end table

2 x minus 1 equals 0 when x equals 1 half:

table row cell straight p open parentheses 1 half close parentheses end cell equals 1 row cell 8 open parentheses 1 half close parentheses to the power of 4 plus a open parentheses 1 half close parentheses squared plus b open parentheses 1 half close parentheses minus 1 end cell equals 1 row cell 1 half plus 1 fourth a plus 1 half b minus 1 end cell equals 1 row cell a plus 2 b end cell equals 6 end table

Solving simultaneously,

table row blank blank cell stack attributes charalign center stackalign right end attributes row a plus b equals 2 end row row a plus 2 b equals 6 end row horizontal line row b equals 4 end row end stack end cell end table

therefore space a equals 2 minus 4 equals negative 2

bold italic a bold equals bold minus bold 2 bold comma bold space bold italic b bold equals bold 4

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