programme f6: polynomial equations worked examples and exercises are in the text stroud programme f6...
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Programme F6: Polynomial equations Worked examples and exercises are in the text STROUD Polynomial equations Quadratic equations Solution of cubic equations having at least one linear factor Solution of fourth-order equations having at least two linear factorsTRANSCRIPT
Programme F6: Polynomial equations
Worked examples and exercises are in the textSTROUD
PROGRAMME F6
POLYNOMIAL EQUATIONS
Programme F6: Polynomial equations
Worked examples and exercises are in the textSTROUD
Polynomial equations
Quadratic equations
Solution of cubic equations having at least one linear factor
Solution of fourth-order equations having at least two linear factors
Programme F6: Polynomial equations
Worked examples and exercises are in the textSTROUD
Polynomial equations
Quadratic equations
Solution of cubic equations having at least one linear factor
Solution of fourth-order equations having at least two linear factors
Programme F6: Polynomial equations
Worked examples and exercises are in the textSTROUD
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.
Programme F6: Polynomial equations
Worked examples and exercises are in the textSTROUD
Polynomial equations
Quadratic equations
Solution of cubic equations having at least one linear factor
Solution of fourth-order equations having at least two linear factors
Programme F6: Polynomial equations
Worked examples and exercises are in the textSTROUD
Polynomial equations
Quadratic equations
Solution of cubic equations having at least one linear factor
Solution of fourth-order equations having at least two linear factors
Programme F6: Polynomial equations
Worked examples and exercises are in the textSTROUD
Quadratic equations, ax2 + bx + c = 0
Solution by factors
Solution by completing the square
Solution by formula
Programme F6: Polynomial equations
Worked examples and exercises are in the textSTROUD
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
Programme F6: Polynomial equations
Worked examples and exercises are in the textSTROUD
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 = 6.606 or x = −0.606 to 3 dp
Programme F6: Polynomial equations
Worked examples and exercises are in the textSTROUD
Quadratic equations, ax2 + bx + c = 0
Solution by formula
To solve ax2 + bx + c = 0 use can be made of the formula:
2 42
b b acx a
Programme F6: Polynomial equations
Worked examples and exercises are in the textSTROUD
Polynomial equations
Quadratic equations
Solution of cubic equations having at least one linear factor
Solution of fourth-order equations having at least two linear factors
Programme F6: Polynomial equations
Worked examples and exercises are in the textSTROUD
Polynomial equations
Quadratic equations
Solution of cubic equations having at least one linear factor
Solution of fourth-order equations having at least two linear factors
Programme F6: Polynomial equations
Worked examples and exercises are in the textSTROUD
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.
Programme F6: Polynomial equations
Worked examples and exercises are in the textSTROUD
Polynomial equations
Quadratic equations
Solution of cubic equations having at least one linear factor
Solution of fourth-order equations having at least two linear factors
Programme F6: Polynomial equations
Worked examples and exercises are in the textSTROUD
Polynomial equations
Quadratic equations
Solution of cubic equations having at least one linear factor
Solution of fourth-order equations having at least two linear factors
Programme F6: Polynomial equations
Worked examples and exercises are in the textSTROUD
Solution of fourth-order equations having at least two linear factors
In Programme F3 fourth-order 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 fourth-order equations.
Programme F6: Polynomial equations
Worked examples and exercises are in the textSTROUD
Learning outcomes
Solve quadratic equations by factors, completing the square and by formula
Solve cubic equations with at least one linear factor
Solve fourth-order equations with at least two linear factors