Factor Theorem (AQA GCSE Further Maths): Revision Note

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:

  • $x^3+4x^2-3x+1$ (degree 3)

  • $5x^6-2x^2$ (degree 6)

  • $10$ (degree 0)

• The following are not polynomials:

  • $\frac{1}{x}+x^3$ (the $\frac{1}{x}$can be written $x^{-1}$ which has a negative power)

  • $x^2+3x+\sqrt{x}$ (the $\sqrt{x}$can be written $x^{\frac{1}{2}}$which has a non-integer power)

  • $x^5+2-\sin x$ ($\sin x$ 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 $6x^4-3x^2+2x+8$

  • ... 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, $\left(x^3+4x\right)+\left(x^2+8x+3\right)$ gives $x^3+x^2+12x+3$

  • whereas $\left(x^3+4x\right)-\left(x^2+8x+3\right)$ gives $x^3-x^2-4x-3$

    • the second bracket expands to $-x^2-8x-3$

• 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, $\left(x^3+4x\right)\left(x^2+8x+3\right)$ can be done as follows:

  •  

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

  • add any like terms that arise: $3x^3+4x^3=7x^3$

  • the final answer is therefore $x^5+8x^4+7x^3+32x^2+12x$

Factor Theorem

What is a factor of a polynomial?

• You already know that some quadratic expressions can be factorised

  • $x^2+5x+6$ factorises to $\left(x+2\right)\left(x+3\right)$

  • (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

• $4x^3+8x^2-9x-18$ factorises to $\left(x+2\right)\left(2x+3\right)\left(2x-3\right)$

  • there are three linear factors
    

    • or, by expanding the last two brackets, $\left(x+2\right)\left(4x^2-9\right)$, 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 $\mathrm{f}\left(x\right)=x^3-6x^2+11x-6$

  • $\mathrm{f}\left(2\right)=2^3-6\times 2^2+11\times 2-6=0$ (zero)

  • $\mathrm{f}\left(4\right)=4^3-6\times 4^2+11\times 4-6=6$ (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 $\mathbf{f}\mathbf{\left(}\frac{\mathit{b}}{\mathit{a}}\mathbf{\right)}\mathbf{=}\mathbf{0}$ then (ax - b) is a factor  

  • It also works in reverse, so if (ax - b) is a factor then $\mathrm{f}\left(\frac{b}{a}\right)=0$

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

• For example, you can show that (2x - 3) is a factor of $\mathrm{f}\left(x\right)=4x^3+8x^2-9x-18$ without doing any factorising

  • If (2x - 3) really is a factor, then the Factor Theorem says $\mathrm{f}\left(\frac{3}{2}\right)$ should equal zero - check to see if that's true

    • $\mathrm{f}\left(\frac{3}{2}\right)=4{\left(\frac{3}{2}\right)}^3+8{\left(\frac{3}{2}\right)}^2-9\left(\frac{3}{2}\right)-18=0$ so yes, (2x - 3) is a factor (by the Factor Theorem)

• Be careful with the signs and fraction order

  •  $\mathrm{f}\left(\frac{3}{2}\right)=0$ means (2x - 3) is a factor, not any of (3x + 2), (3x - 2) or (2x + 3)