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Polynomial | Remainder Theorem | Factor Theorem 3/3

C. SOLUTION OF EQUATIONS

Question: Solve 3x3 – 10x2 + 9x – 2 = 0

When you spot the keywords 

– “Solve” and

– equation = 0 

– “Find the roots of equation” 

in your exam question, it means you’ll have to:

  • Find the factors of an equation f(x) (usually cubic)
  • Find the roots of f(x)=0 (i.e. final answer must be in the form: x = a, b … where a, b, … are the roots)

Ans: Let f(x) = 3x3 – 10x2 + 9x – 2

Find the first factor via trial and error

Try x=1: f(1) = 3(1)3 – 10(1)2 + 9(1) – 2 = 0
⇒ (x-1) is a factor

Find the remainder expression by either LONG DIVISION or COMPARING COEFFICIENTs

LONG DIVISION METHOD
(so short, i like it)

Comparing Coefficient Method

Form the expression after factorising f(x) with the 1st factor found

3x3– 10x2+9x-2 ≡ (x-1)(ax2+bx+c)

Compare coefficients of highest power and constant first

Compare x3:

3x3 = ax3  –> a=3

Compare constant:

-2= -c  –> c=2

Then compare coefficients of middle powers

Compare x

9x=cx-bx

Substitute c=2

9x=2x-bx

b=-7

so f(x)= (x-1)(3x2-7x+2) 🙂

But Question wants us to find all the roots, solve for all x, so we have to further solve (3x2-7x+2) , there are 4 methods for you to choose from, but i will use the frame method.

f(x)=(x-1)(x-2)(3x-1)

so x-1=0, x-2=0, 3x-1=0


so x=1 or 2 or 1/3 🙂

Differentiate between question asking you to

  • Factorize –> f(x)=(x-1)(x-2)(3x-1)
  • Solve –> x=1 or 2 or 1/3 🙂

If the quadratic equation in Step 3 cannot be easily factorized → you’ll have to use the method 4 in Quadratic Formula to find the two solutions. You’ll normally need to use method 4 when you see terms like ± or
surds within the question or when the question ask you to leave the answer in exact form.

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