1. PROGRAMME F6
POLYNOMIAL EQUATIONS

2. Programme F6: Polynomial equations
Quadratic equations
Solution of cubic equations having at least one linear factor
Solution of quartic equations having at least two linear factors

3. Programme F6: Polynomial equations
Quadratic equations
Solution of cubic equations having at least one linear factor
Solution of quartic equations having at least two linear factors

4. Programme F6: Polynomial equations
In Programme F3 a polynomial in the variable x was evaluated by substituting the x-value into the equation and finding the resulting value for the polynomial expression.
This process is known as evaluating the expression.
Here the process is reversed by giving the polynomial expression the value of zero and finding those values of x which satisfy the resulting equation.

5. Programme F6: Polynomial equations
Quadratic equations
Solution of cubic equations having at least one linear factor
Solution of quartic equations having at least two linear factors

6. Programme F6: Polynomial equations
Quadratic equations, ax2 + bx + c = 0
Solution by factors
Solution by completing the square
Solution by formula

7. Programme F6: Polynomial equations
Quadratic equations, ax2 + bx + c = 0
Solution by factors
Where simple factors exist the solution can be derived from those. For example:
x2 + 5x – 14 can be factorized as (x + 7)(x – 2)
so if:
x2 + 5x – 14 = 0 then (x + 7)(x – 2) = 0 and so x = −7 or x = 2

8. x2 – 6x + (–3)2 = 4 + (–3)2 that is x2 – 6x + 9 = (x – 3)2 = 13
Programme F6: Polynomial equations
Quadratic equations, ax2 + bx + c = 0
Solution by completing the square
Where simple factors do not exist the solution can be derived from completing the square. For example to solve x2 – 6x – 4 = 0 it is noted that x2 – 6x – 4 does not have simple factors so add 4 to both sides to give:
x2 – 6x = 4
Now, add the square of half the x-coefficient to both sides to give:
x2 – 6x + (–3)2 = 4 + (–3)2 that is x2 – 6x + 9 = (x – 3)2 = 13
Therefore x – 3 = ±√13 so x = or x = −0.606 to 3 dp

9. Programme F6: Polynomial equations
Quadratic equations, ax2 + bx + c = 0
Solution by formula
To solve ax2 + bx + c = 0 use can be made of the formula:

10. Programme F6: Polynomial equations
Quadratic equations
Solution of cubic equations having at least one linear factor
Solution of quartic equations having at least two linear factors

11. Programme F6: Polynomial equations
Solution of cubic equations having at least one linear factor
In Programme F3 cubic polynomials were factorized with application of the remainder theorem and the factor theorem and the evaluation of polynomials by nesting.
These methods are reapplied to solve cubic equations.

12. Programme F6: Polynomial equations
Quadratic equations
Solution of cubic equations having at least one linear factor
Solution of quartic equations having at least two linear factors

13. Programme F6: Polynomial equations
Solution of quartic equations having at least two linear factors
In Programme F3 quartic polynomials were factorized with application of the remainder theorem and the factor theorem and the evaluation of polynomials by nesting.
These methods are reapplied to solve quartic equations.

14. Programme F6: Polynomial equations
Learning outcomes
Solve quadratic equations by factors, completing the square and by formula
Solve cubic equations with at least one linear factor
Solve quartic equations with at least two linear factors