5.2 Q-5

Exercise 5.2 — Question 5

This question applies the Factor Theorem and hit-and-trial method to fully factorize cubic polynomials.

---

Key Theorems

Remainder Theorem: If a polynomial $P(x)$ is divided by $(x-a)$, the remainder is $P(a)$.

Factor Theorem: $(x-a)$ is a factor of $P(x)$ if and only if $P(a)=0$.

---

Method: Factorizing a Cubic Polynomial

To factorize $P(x)=a{x}^3+b{x}^2+cx+d$:

1. Find a rational zero using hit-and-trial: test factors of the constant term $d$ (divided by factors of leading coefficient $a$).
2. Confirm the zero by substituting into $P(x)$ — if $P(k)=0$, then $(x-k)$ is a factor.
3. Divide $P(x)$ by $(x-k)$ using synthetic division or long division to get a quadratic quotient $Q(x)$.
4. Factorize the quadratic $Q(x)$ by factoring or the quadratic formula.
5. Write the complete factorization: $P(x)=(x-k)\cdot Q(x)$.

---

Worked Example

Factorize $P(x)=x^3-6x^2+11x-6$.

Step 1: Factors of constant term $-6$: $\pm 1,\pm 2,\pm 3,\pm 6$.

Step 2: Test $x=1$: $P(1)=1-6+11-6=0✓$ So $(x-1)$ is a factor.

Step 3: Divide by $(x-1)$ using synthetic division: $x^3-6x^2+11x-6=(x-1)(x^2-5x+6)$

Step 4: Factorize $x^2-5x+6$: $x^2-5x+6=(x-2)(x-3)$

Step 5: Complete factorization: $P(x)=(x-1)(x-2)(x-3)$

---

Synthetic Division Reminder

To divide $P(x)$ by $(x-k)$, write the coefficients of $P(x)$ and apply:

$\begin{array}{rrrrr}k& a_n& a_{n-1}& \cdots & a_0\\ & & k\cdot b_n& \cdots & \\ & b_n& b_{n-1}& \cdots & R\end{array}$

where $R=P(k)$ is the remainder.

---