Q4(i):

Factorise : (i) $12x^2 – 7x + 1$

Initial Setup & Polynomial Identification

We are tasked with factorising the quadratic polynomial $P(x) = 12x^2 - 7x + 1$. [Per the Fundamental Theorem of Algebra and the properties of quadratic expressions, a polynomial of degree 2 can generally be factored into the product of two linear binomials]. We will utilize the method of splitting the middle term (also known as the AC method).

Step 1: Determine the Coefficients for the AC Method

A standard quadratic polynomial is expressed in the form $ax^2 + bx + c$. By comparing our given polynomial to the standard form, we identify the coefficients:

• $a = 12$ (Coefficient of $x^2$)
• $b = -7$ (Coefficient of $x$)
• $c = 1$ (Constant term)

Step 2: Find the Splitting Factors

To split the middle term, we must find two integers, let us define them as $p$ and $q$, that satisfy two specific conditions simultaneously:

1. Product Condition: $p \times q = a \times c = 12 \times 1 = 12$
2. Sum Condition: $p + q = b = -7$

[This algebraic condition ensures that the middle term is partitioned in a way that maintains the polynomial's equivalence while allowing for factorisation by grouping].

We list the factor pairs of $12$: $(1, 12)$, $(2, 6)$, and $(3, 4)$. Because the product ($+12$) is positive and the sum ($-7$) is negative, both factors $p$ and $q$ must be negative integers. Let us test the negative pairs:

• $(-1) + (-12) = -13$ (Incorrect)
• $(-2) + (-6) = -8$ (Incorrect)
• $(-3) + (-4) = -7$ (Correct)

Thus, the required splitting factors are $p = -4$ and $q = -3$.

Step 3: Rewrite the Polynomial by Splitting the Middle Term

We substitute the middle term $-7x$ with the equivalent expression $-4x - 3x$:

$P(x) = 12x^2 - 4x - 3x + 1$

Step 4: Factorise by Grouping

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

$(12x^2 - 4x) - (3x - 1)$

Analyzing the first group: $(12x^2 - 4x)$
The highest common numerical factor of $12$ and $4$ is $4$. The highest common variable factor of $x^2$ and $x$ is $x$. Therefore, the GCF is $4x$.
Factoring out $4x$ yields: $4x(3x - 1)$

Analyzing the second group: $-(3x - 1)$
To ensure the binomial inside the parentheses matches the first group, we factor out $-1$.
Factoring out $-1$ yields: $-1(3x - 1)$

[By the Distributive Property of Multiplication over Addition, $ab + ac = a(b+c)$. We apply this in reverse to factor out the GCF].

Step 5: Extract the Common Binomial Factor

Substitute the factored groups back into the main expression:

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

Observe that the binomial $(3x - 1)$ is now a common factor to both major terms. We factor out $(3x - 1)$ from the entire expression:

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

Final Solution: The factorised form of the polynomial $12x^2 - 7x + 1$ is $(4x - 1)(3x - 1)$.