Q4(ii):

Factorise : (ii) $2x^2 + 7x + 3$

Initial Setup & Theoretical Foundation

We are tasked with factorising the quadratic polynomial:

$P(x) = 2x^2 + 7x + 3$

To factorise a quadratic polynomial of the standard form $ax^2 + bx + c$, we employ the "Splitting the Middle Term" method (also known as the AC method). This technique relies on finding two integers, $p$ and $q$, that satisfy two specific conditions simultaneously:

• Sum Condition: $p + q = b$ [The coefficient of the middle term]
• Product Condition: $p \times q = a \times c$ [The product of the leading coefficient and the constant term]

Step 1: Identifying Coefficients for the AC Method

By comparing the given polynomial $2x^2 + 7x + 3$ with the standard quadratic form $ax^2 + bx + c$, we extract the following coefficients:

• Leading coefficient ($a$) = $2$
• Middle coefficient ($b$) = $7$
• Constant term ($c$) = $3$

Next, we calculate the target product ($ac$):

$a \times c = 2 \times 3 = 6$

Step 2: Determining the Splitting Factors

We must find two integers $p$ and $q$ such that:

$p + q = 7$

$p \times q = 6$

Let us systematically evaluate the factor pairs of $6$:

The correct integers are $6$ and $1$.

Step 3: Rewriting the Polynomial

We substitute the middle term $7x$ with the sum of $6x$ and $1x$ [Per the distributive property, $7x = (6 + 1)x = 6x + x$]:

$2x^2 + 6x + x + 3$

Step 4: Factorisation by Grouping

We now group the four terms into two pairs to extract the Greatest Common Factor (GCF) from each pair:

$= (2x^2 + 6x) + (x + 3)$

Extracting the GCF from the first group: The terms $2x^2$ and $6x$ share a common factor of $2x$.

$2x(x + 3)$

Extracting the GCF from the second group: The terms $x$ and $3$ share no common factors other than $1$.

$+ 1(x + 3)$

Reassembling the expression yields:

$= 2x(x + 3) + 1(x + 3)$

Notice that the binomial $(x + 3)$ is now a common factor to both major terms. We factor out $(x + 3)$ [By the Distributive Property of Multiplication over Addition, $AB + CB = (A + C)B$]:

$= (x + 3)(2x + 1)$

Visual Representation: Area Model of Factorisation

The area model geometrically validates our algebraic factorisation. The total area of the rectangle represents the polynomial $2x^2 + 7x + 3$, while the side lengths represent the factors $(2x + 1)$ and $(x + 3)$.

Final Solution: The factorised form of the polynomial $2x^2 + 7x + 3$ is $(x + 3)(2x + 1)$.