Q4(iii):

Factorise : (iii) $6x^2 + 5x – 6$

Initial Setup & Given Polynomial

We are tasked with factorising the quadratic polynomial:

$P(x) = 6x^2 + 5x - 6$

This expression is in the standard quadratic form $ax^2 + bx + c$. To factorise it, we will employ the Splitting the Middle Term technique, also known as the AC Method [Per the fundamental principles of polynomial factorisation over integers].

Step 1: Coefficient Analysis & The AC Method

First, we identify the coefficients of the quadratic polynomial:

• Leading coefficient ($a$) = $6$
• Middle coefficient ($b$) = $5$
• Constant term ($c$) = $-6$

According to the AC Method, we must find the product of the leading coefficient and the constant term ($a \times c$):

$a \times c = 6 \times (-6) = -36$

Step 2: Splitting the Middle Term

We must now find two integers, let's call them $p$ and $q$, that satisfy two conditions simultaneously:

1. Their product must equal $ac$: $p \times q = -36$
2. Their sum must equal $b$: $p + q = 5$

Since the product ($-36$) is negative, the two numbers must have opposite signs. Since their sum ($5$) is positive, the number with the larger absolute value must be positive. Let us systematically evaluate the factor pairs of $36$:

The correct integers are $9$ and $-4$. We will use these to split the middle term ($5x$).

Step 3: Algebraic Substitution and Grouping

Substitute $5x$ with $9x - 4x$ in the original polynomial:

$6x^2 + 9x - 4x - 6$

Next, we group the terms into pairs to factor out the Greatest Common Divisor (GCD) from each pair [Applying the Distributive Property $ab + ac = a(b+c)$]:

$= (6x^2 + 9x) - (4x + 6)$

Extract the GCD from the first group ($6x^2 + 9x$). The GCD of $6$ and $9$ is $3$, and the GCD of $x^2$ and $x$ is $x$. Thus, we factor out $3x$:

$= 3x(2x + 3) - (4x + 6)$

Extract the GCD from the second group ($4x + 6$). The GCD of $4$ and $6$ is $2$. To ensure the binomial inside the parentheses matches the first group, we factor out $-2$:

$= 3x(2x + 3) - 2(2x + 3)$

Step 4: Factoring out the Common Binomial

Notice that the binomial $(2x + 3)$ is now a common factor in both terms. We factor it out [By the reverse distributive property]:

$= (2x + 3)(3x - 2)$

Geometric Verification: Area Model

To rigorously verify our algebraic manipulation, we can map the grouped terms to a geometric area model. The total area of the rectangle represents the polynomial $6x^2 + 5x - 6$, while the side lengths represent its factors $(2x + 3)$ and $(3x - 2)$.

Final Solution: The factorised form of the polynomial $6x^2 + 5x - 6$ is $(2x + 3)(3x - 2)$.