CAT Concept of splitting middle term - Practice Questions & MCQ

Definition: 

Splitting the middle term is a method primarily used for factoring quadratic polynomials. In this technique, the middle term (the term containing x to the power 1 in a quadratic polynomial) is split into two terms in such a way that factoring becomes possible.

Procedure:

For a quadratic polynomial of the form 

1. Multiply a (coefficient of  and c (constant term).

2. Find two numbers whose sum is b (the coefficient of x) and the product is ac (from the previous step).

3. Rewrite the middle term (bx) using the two numbers found in step 2.

4. Group the terms in pairs and factor out the common factors.

Application of Previous Concepts: 

Understanding the degree and the type (whether it's a monomial, binomial, or trinomial) of polynomial is vital before attempting to split the middle term. 

Foundation Building Questions:

Question 1: 

Factorise the polynomial .

Solution: 

1. Multiply the coefficient of x2 and the constant: 6 x (-6) = -36

2. The numbers whose sum is 5 and product is -36 are 9 and -4.

3. Rewrite the middle term: 6x2 + 9x - 4x - 6

4. Group the terms: 3x(2x + 3) - 2(2x + 3)

5. Factor out the common factor: (2x + 3)(3x - 2)

So, P(x) = (2x + 3)(3x - 2).

Question 2: 

Factorise the polynomial 

Solution: 

1. Multiply the coefficient of x2 (which is 1 here) and the constant: 1 x 12 = 12

2. Numbers that sum up to -7 and multiply to give 12 are -4 and -3.

3. Rewrite the middle term: 

4. Group the terms: x(x - 4) - 3(x - 4)

5. Factor out the common factor: (x - 4)(x - 3)

So, Q(x) = (x - 4)(x - 3).

Tips and Tricks:

1. Product AC: Finding the product of a and c (we often call it 'product AC' in this method) is the key step. If you get this step right, the rest usually falls into place.

2. Common Factor First: Before trying to split the middle term, always check if there's a common factor you can factor out from all terms. This often simplifies the problem.

3. Practice: This method, more than others, benefits greatly from practice. The more you practise, the quicker you'll get at spotting the numbers needed to split the middle term.

4. Degree Awareness: This method is mainly for quadratic polynomials. If you're given a higher-degree polynomial, consider other techniques or reduce its degree (if possible) before attempting this method.

Splitting the middle term is a foundational algebraic skill, especially for quadratic equations. It's particularly useful in problems where quadratic expressions need to be factorised or simplified.