Factor Theorem (AQA GCSE Further Maths)

Definition of a Polynomial

What is a polynomial?

• A polynomial is a sum of terms with non-negative integer powers of x 
  

  • the highest power of x is its degree

• The following are polynomials:

  • (degree 3)

  • (degree 6)

  • (degree 0)

• The following are not polynomials:

  • (the can be written  which has a negative power)

  • (the can be written which has a non-integer power)

  • ( is not a power of x)

• There are words to name different types of polynomials:

  • degree	name
    0	constant
    1	linear
    2	quadratic
    3	cubic
    4	quartic
    5	quintic
    ...	...

• For the polynomial

  • ... the degree is 4 (it's a quartic)

  • ... the coefficient of x² is -3

  • ... the quadratic term (or term involving x²) is -3x²

  • ... the coefficient of x³ is 0

  • ... the constant is 8

How do I add / subtract / multiply polynomials?

• Adding and subtracting two polynomials requires collecting like terms

  • for example,  gives 

  • whereas gives 

    • the second bracket expands to 

• Multiplying two polynomials can be done by expanding brackets or using a grid method (multiplying rows by columns then combining terms to get the answer)

  • for example,  can be done as follows:

  •  

    	x3	4x
    x2	x5	4x3
    8x	8x4	32x2
    3	3x3	12x

  • add any like terms that arise:

  • the final answer is therefore

Factor Theorem

What is a factor of a polynomial?

• You already know that some quadratic expressions can be factorised

  • factorises to 

  • (x + 2) and (x + 3) are called factors

    • as the power of x in each factor is 1, they can also be called linear factors

• Similarly, other polynomials can be factorised

• factorises to 

  • there are three linear factors
    

    • or, by expanding the last two brackets, , you could write it as one linear factor and one quadratic factor

• Rational factors refer to linear factors in the form (ax + b), with a number in front of the x, like (2x + 3)

What is the Factor Theorem for (x - a)?

• Let f(x) be a polynomial

  • The Factor Theorem states that if f(a) = 0 then (x - a) is a factor  

  • It also works in reverse, so if (x - a) is a factor then f(a) = 0

• For example, try substituting x = 2 and x = 4 into

  • (zero)

  • (not zero)

  • The Factor Theorem says that (x - 2) is a factor of f(x), but (x - 4) is not

    • It tells you without you having to factorise f(x)

• Be careful with the signs

  • f(2) = 0 means (x - 2) is a factor, not (x + 2)

What is the Factor Theorem for (ax - b)?

• Let f(x) be a polynomial

  • The Factor Theorem above can be extended to say that if  then (ax - b) is a factor  

  • It also works in reverse, so if (ax - b) is a factor then 

• This is sometimes called The Factor Theorem for rational factors, (ax - b)

• For example, you can show that (2x - 3) is a factor of without doing any factorising

  • If (2x - 3) really is a factor, then the Factor Theorem says should equal zero - check to see if that's true

    • so yes, (2x - 3) is a factor (by the Factor Theorem)

• Be careful with the signs and fraction order

  •   means (3x - 2) is a factor, not any of (3x + 2), (2x - 3) or (2x + 3)