Below are several formulas that are useful for Sec 2 students, those marked with a * is additional information and are not frequently seen in Sec 2 syllabus.
Formula for solving roots for quadratic equations
For quadratic equations in the form , it has solutions
The discriminant of the quadratic equation, , determines the properties of the roots.
For , the solutions are complex, in other words, the quadratic equation has no real roots.
For , the solutions degenerate to one real solution, in other words, the quadratic equation has two equal real roots.
For , the solutions are distinct and real, in other words, the quadratic equation has two distinct real roots.
The Binomial Theorem
For , we can derive the formula:
where is the binomial coefficient, satisfying the conditions of , and is defined as
Also, an interesting fact to note is that
This implies that the sum of binomial coefficients of the expression is . It also comes with an interesting proof:
Let in the binomial formula be 1, which leads us to
Since the condition for the binomial formula to be true is and , we can simplify the expression to:
The Factor Theorem
A polynomial is divisible by the binomal if and only if . This is quite self-explanatory, so I won’t elaborate on this. Ok… I change my mind. is a function of P. Usually, is used.
When you put something in a function, what it really means is you are giving a value. So in , all become 0 and whatever is left of the function is the remainder. I will explain more under remainder theorem. In , there is no remainder and therefore (x-a) is a multiple of the polynomial.
The Remainder Theorem
It is also worth noting that , where is a polynomial, is a linear factor, where is just some number. Also, is the quotient polynomial when is divided by the factor , while is the remainder.
That is all under the topic of polynomials, other formulas will be written in detail in the respective future posts.