How to ace grade 12 math

rationalizing denominator conjugate pair

Congratulations! You are in your last year of high school and you want it to be a successful year. If you are planning to continue studying in a post-secondary institution, one of the grade 12 math courses is usually a requirement and you, no doubt, would want to ace grade 12 math.

No matter what grade 12 math course you are taking, there is a set of skills and knowledge that will definitely help you succeed.

Reducing Fractions

Knowing how to reduce fractions will help you get the correct answer and prevent you from losing marks for not expressing your answer in the simplest form.

Not every fraction can be reduced, but if it can, it must be reduced. In order to reduce a fraction, you need to determine the greatest common factor for the numerator and the denominator. Then divide both the numerator and the denominator by that factor and record the resulting fraction.

reducing a fraction

Simplifying Radicals

Another important skill in grade 12 that will most definitely help you earn better grades is simplifying radicals and rationalizing a denominator.

Sometimes we get a value under the square root that is not a perfect square and cannot produce an integer when square rooted. We have to keep it under the square root in order to have an exact answer (once you punch it into a calculator and record it as a decimal, your answer is no longer exact). However, we can always try to split the value under the square root into the product of two numbers one of which would, in fact, be a perfect square and thus could be square rooted, producing an integer.

simplifying radical

It is equally important to remember to rationalize the denominator when the denominator is a radical. When it comes to trigonometric ratios, the ratios for special angles are frequently determined in their exact form, containing a radical. Therefore, knowing how to rationalize the denominator will help ensure that you do not loose marks here and there for not expressing your solutions in the simplest form.

In order to rationalize a denominator that is just one value under the square root, for example, all you have to do is multiply both the denominator and the numerator by that same value under the square root. That will result in an integer in the denominator.

rationalizing denominator

If the denominator is an expression containing a radical, then both the denominator and the numerator need to be multiplied by the conjugate pair of that expression (an expression with the opposite sign).

rationalizing denominator conjugate pair

Factoring Quadratic Expressions

Knowledge of common factoring, factoring simple trinomials and complex trinomials, as well as recognizing special factoring cases such as difference of squares and perfect square trinomials is absolutely crucial for your success in any grade 12 math course.

Let’s consider all of the above mentioned types of factoring so that you know what to study and practice if you feel you need to improve those skills.

Common factoring is the simplest form of factoring polynomial expressions. When factoring quadratic expressions, always look out to see if there is a common factor first, then proceed with other types of factoring where possible.

common factoring quadratics

Factoring a simple trinomial means to factor a quadratic expression with the leading coefficient of 1, by finding the two numbers that multiply to equal the constant and add to equal the coefficient of the middle term in the expression.

factoring simple trinomial

Factoring a complex trinomial means factoring a quadratic expression whose leading coefficient is not 1. In order to factor such trinomial it is necessary to decompose the middle term of the expression into two terms and then group two pairs of the resulting terms, common factoring at the end. Here is how

factoring complex trinomial

There are two special cases of factoring quadratics that frequently come up in grade 12 math in various situations (even when solving trigonometric equations or proving trig identities).

The first one is the difference of squares and here is an example

difference of squares factoring

The second one is the perfect square trinomial and here is how to recognize it and factor it

perfect square trinomials factoring

There are many other things, of course, that you would have to keep in mind and practice in order to ace grade 12 math. However, the above described skills are the most commonly used skills. They are also easy to review and perfect in a short period of time.

So, don’t waste your time, make sure that your are proficient in the above and enjoy your grade 12 math success!

Here is radicals practice

Here is factoring practice


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