Q4(iv):

Factorise : (iv) $3x^2 – x – 4$

Initial Setup & Polynomial Identification

We are tasked with factorising the given quadratic polynomial:

$P(x) = 3x^2 - x - 4$

[Per the Fundamental Theorem of Algebra and the structure of polynomials], a quadratic polynomial in a single variable is expressed in the standard form $ax^2 + bx + c$, where $a$, $b$, and $c$ are real constants, and $a \neq 0$. By comparing our given polynomial to the standard form, we can identify the coefficients:

• Leading coefficient ($a$) = $3$
• Linear coefficient ($b$) = $-1$
• Constant term ($c$) = $-4$

Step 1: The AC Method & Coefficient Analysis

To factorise a quadratic polynomial of the form $ax^2 + bx + c$ over the integers, we employ the method of splitting the middle term (also known as the AC Method). This requires us to find two integers, let us call them $p$ and $q$, that satisfy two simultaneous conditions:

1. The product of the two integers must equal the product of the leading coefficient and the constant term: $p \times q = a \times c$.
2. The sum of the two integers must equal the linear coefficient: $p + q = b$.

Calculating the required product ($a \times c$):

$a \times c = (3) \times (-4) = -12$

The required sum ($b$) is $-1$.

Step 2: Determining the Factors

We must systematically evaluate the integer factor pairs of $-12$ to find the specific pair that sums to $-1$.

The integers that satisfy both conditions are $3$ and $-4$.

Step 3: Splitting the Middle Term

We now rewrite the original linear term ($-x$) as the sum of the two terms derived from our factors: $3x$ and $-4x$.

$P(x) = 3x^2 + 3x - 4x - 4$

Step 4: Factorisation by Grouping

[By the Distributive Property of Multiplication over Addition], we can group the polynomial into two binomial pairs and extract the greatest common factor (GCF) from each pair.

Group the terms:

$(3x^2 + 3x) - (4x + 4)$

Extract the GCF from the first group ($3x^2 + 3x$). The GCF is $3x$:

$3x(x + 1)$

Extract the GCF from the second group ($-4x - 4$). To ensure the binomial inside the parentheses matches the first group, we extract $-4$:

$-4(x + 1)$

Substitute these back into the expression:

$3x(x + 1) - 4(x + 1)$

Notice that $(x + 1)$ is now a common binomial factor. We factor out $(x + 1)$:

$(x + 1)(3x - 4)$

Visual Representation: Area Model of Factorisation

The algebraic grouping can be geometrically verified using an area model. The total area of the rectangle represents the quadratic polynomial $3x^2 - x - 4$, while the dimensions (length and width) represent its linear factors $(3x - 4)$ and $(x + 1)$.

Final Verification

To ensure absolute rigor, we expand the factored form to verify it yields the original polynomial:

$(x + 1)(3x - 4) = x(3x - 4) + 1(3x - 4)$

$= 3x^2 - 4x + 3x - 4$

$= 3x^2 - x - 4$

The expansion perfectly matches the initial polynomial, confirming the accuracy of the factorisation.

Final Solution: The factorised form of the polynomial $3x^2 - x - 4$ is $(x + 1)(3x - 4)$.