## Improper fraction

Improper fraction can be expressed in proper fraction without doing long division.

## Try this long division problem!

Many students, including JC H2 math students, have difficulty doing long division when it comes to complex problems.

Try this problem to test or sharpen your long division skills.

Question

If  is a factor of , prove by long division that

Solution

## Synthetic division

Synthetic division can be used to divide a polynomial by a linear divisor, which is commonly required in H2 math and Amath.

It is faster than the traditional long division.

How to do Synthetic division

## Partial fraction: Cover-up rule

Partial fraction is commonly used in binomial expansion, method of difference and integration.

When the divisor consists only of linear factors, cover up rule can be used to find the constants in the partial fraction quickly.

Example

Express  in partial fraction.

By cover up rule,

Therefore

0 = A+C

Therefore,

## Inequalities: Use And/OR correctly

Many students use the words “And/Or” indiscriminately in solving inequalities. However, there is a very important difference. ‘And‘ means you take the intersection of the solutions sets. ‘Or‘ means you take the union of the solutions sets.

Example

## Not being offered Amath in school? Don’t give up yet

Some schools require their students to score at least 75% in their Sec 2 end of year exam, before they are allowed to take additional maths in Sec 3.

For students who are not offered Amath in school and still want to take Amath in O level, they can get the following textbook and assessment book, and self study on their own or find a private tutor to teach them.

Talk to the school and request that the school help you register Amath in O level. The school will probably require you to pass sec 3 end of year examination before they allow you to take Amath at O level.

I have a student who was not offered Amath in school. She wanted to take Amath at O level so she could have more options in future. Her school only allowed her to take the subject on condition that she passed her Sec 3 End of Year A Math exams. The student was not given any opportunity by the school to attend A Math classes. I tutored her and within a short span of 3.5 months, the student had obtained 64 marks (high B4), better than her cohort average of 60.

Though the school now allow her to take Amath at O level, she still cannot take Amath lesson in school, because it clashed with her Principle of Accounts. So I continue to teach her Amath and finished the entire O level syllabus in another 3 months. Eventually she scored A1 in her school sec 4 mid year exams and B3 in O level.

So don’t give up if you really wish to do Amath 🙂

## Understanding R formula

Quite a number of students have difficulty understanding R formula.

Hopefully this graph will help.

3 sin(x)+4 cos (x) expressed in R formula is 5 sin(x+0.927)

So the two graphs appear as 1 graph.

Max value of 3 sin(x)+4 cos (x) = 5. Occurs when sin(x+0.927)=1

Min value of 3 sin(x)+4 cos (x) = -5. Occurs when sin(x+0.927)=-1

Students can try their hands in R formula at https://www.desmos.com/calculator

## Find the area of shaded region

Suppose you are required to find the area of the shaded region.

In Amath, students have yet to learn how to integrate ln x. So is there another way to find the area?

Easier way is to find the yellow area by integrating  $e^y$ with respect to y and then subtract this area from the rectangle.

Creativity helps in solving maths 🙂