Example 1. Find the zero of the polynomial $p(x) = 3x + 1$
Solution $p(x) = 3x + 1$
A zero of a polynomial is a number which makes $p(x) = 0$

Now $p(x) = 0$ ⇒ $3x + 1 = 0$ ⇒ $3x = -1$ ⇒ $x = \frac{-1}{3}$

∴ The zero of the given polynomial $p(x)$ is $\frac{-1}{3}$


Example 2. Find the remainder when $3x^4 - 6x^2 - 8x + 2$ is divided by $x - 2$
Solution Let $p(x) = 3x^4 - 6 x^2 -8x + 2$
$x - 2$ ⇒ $x = 2$
By the Remainder Theorem, we know that when a polynomial is divided by $x - 2$, the remainder is $p(2)$

$\begin{align*} & ∴ p(2) = 3*2^4 - 6*2^2 - 8*2 + 2 \\ & = 3 * 16 - 6 * 4 - 16 + 2 \\ & = 48 - 24 - 16 + 2 \\ & = 10 \end{align*} $


Example 3. The polynomials $(ax^3 + 3x^2 - 3)$ and $(2x^3 - 5x + a)$ when divided by $x - 4$ leave the same remainder. Find the value of $a$.
Solution Let $p(x) = ax^3 + 3x^2 - 3$ and $q(x) = 2x^3 - 5x + a$
$x - 4$ ⇒ $x = 4$
By the Remainder Theorem, we know that when a polynomial is divided by $x - 4$, the remainder is $p(4)$

$\begin{align*} & ∴ p(4) = a*4^3 + 3*4^2 - 3 \\ & = a * 64 + 3 * 16 - 3 \\ & = 64a + 45 \end{align*} $

$\begin{align*} & ∴ q(4) = 2*4^3 - 5*4 + a \\ & = 2 * 64 - 20 + a \\ & = 108 + a \end{align*} $

As the remainder in both the above cases is the same
$ 64a + 45 = 108 + a$
⇒ $63a = 63$
⇒ $a = 1$


Example 4. Factorize $8a^3 + b^3 + 12a^2b + 6ab^2$
Solution To factorize this expression, we will use the Polynomial Identity :
$(x + y)^3 = x^3 + y^3 + 3xy(x + y)$

The given expression can be written as
$\begin{align*} & 8a^3 + b^3 + 12a^2b + 6ab^2 \\ & = (2a)^3 + (b)^3 + 6ab(2a) + 6ab(b) \\ & = (2a)^3 + (b)^3 + 6ab(2a + b) \\ & = (2a + b)^3 \end{align*} $


Example 5. Factorize $64a^3 - 27b^3 - 144a^2b + 108ab^2$
Solution To factorize this expression, we will use the Polynomial Identity :
$(x - y)^3 = x^3 - y^3 - 3xy(x - y)$

The given expression can be written as
$\begin{align*} & 64a^3 - 27b^3 - 144a^2b + 108ab^2 \\ & = (4a)^3 - (3b)^3 - 12ab(12a) + 12ab(9b) \\ & = (4a)^3 + (3b)^3 - 12ab(12a - 9b) \\ & = (4a - 3b)^3 \end{align*} $


Example 6. Factorize $x^4 - 625$
Solution To factorize this expression, we will use the Polynomial Identity :
$(x^2 - y^2) = (x + y)(x - y)$

The given expression can be written as
$\begin{align*} & x^4 - 625 \\ & = (x^2 - 25)(x^2 + 25) \\ & = (x - 5)(x + 5)(x^2 + 25) \end{align*} $


Example 7. Give possible expressions for the length and breadth of the following rectangle, in which its area is given:

Solution Area of a rectangle is the product of its length and breadth. Hence to find possible expressions of the length and the breadth, we need to find factors of this expression.

Comparing the given expression, we observe that, if we put $5a$ as $x$, the given expression can be re-written as
$x^2 - 7x + 12$

To factorize this expression, we will use the Polynomial Identity :
$(x + a)(x + b) = x^2 + (a + b)x + ab$

Thus $-7$ can be split as $-4$ and $-3$

$\begin{align*} & x^2 - 7x + 12 \\ & = x^2 - 4x -3x + 12 \\ & = x(x - 4) -3(x - 4) \\ & = (x - 4)(x - 3) \end{align*} $

Replace $x$ by $5a$ to get back the factors as $(5a - 4)$ and $(5a - 3)$
Therefore, possible expressions for length and breadth are $(5a - 4)$ and $(5a - 3)$


Example 8. Expand $(3x + 2)^3$
Solution To expand this expression, we will use the Polynomial Identity :
$(x + y)^3 = x^3 + y^3 + 3xy(x + y)$

$\begin{align*} & (3x + 2)^3 \\ & = 27x^3 + 8 + 3*3x*2(3x + 2) \\ & = 27x^3 + 8 + 18x(3x + 2) \\ & = 27x^3 + 8 + 54x^2 + 36x \end{align*} $


Example 9. Factorize $(a + b)^3 - 8$
Solution To expand this expression, we will use the Polynomial Identity :
$(x^3 - y^3) = (x - y)(x^2 + xy + y^2)$

$\begin{align*} & (a + b)^3 - 8 \\ & = (a + b - 2)((a + b)^2 + (a + b)*2 + 2^2) \\ & = (a + b - 2)((a + b)^2 + 2(a + b) + 4) \end{align*} $


Example 10. Factorize $(a + b)^3 - (a - b)^3$
Solution To expand this expression, we will use the Polynomial Identity :
$(x^3 - y^3) = (x - y)(x^2 + xy + y^2)$

$\begin{align*} & (a + b)^3 - (a - b)^3 \\ & = ((a + b) - (a - b))((a + b)^2 + (a + b)*(a - b) + (a - b)^2) \\ & = (a + b - a + b)(a^2 + 2ab + b^2 + a^2 - b^2 + a^2 - 2ab + b^2) \\ & = (2b)(a^2 + a^2 + a^2 + b^2) \\ & = (2b)(3a^2 + b^2) \\ \end{align*} $


Example 11. Factorize $a^3 - \frac{1}{a^3} - 2a + \frac{2}{a}$
Solution To expand this expression, we will use the following Polynomial Identity :
$(x^3 - y^3) = (x - y)(x^2 + xy + y^2)$

$\begin{align*} & a^3 - \frac{1}{a^3} - 2a + \frac{2}{a} \\ & = \left(a - \frac{1}{a}\right)\left(a^2 + a*\frac{1}{a} + \frac{1}{a^2}\right) - 2a + \frac{2}{a} \\ & = \left(a - \frac{1}{a}\right)\left(a^2 + 1 + \frac{1}{a^2}\right) - 2\left(a - \frac{1}{a}\right) \\ & = \left(a - \frac{1}{a}\right)\left(a^2 + 1 + \frac{1}{a^2} - 2\right) \\ & = \left(a - \frac{1}{a}\right)\left(a^2 - 1 + \frac{1}{a^2}\right) \\ \end{align*} $


Example 12. Factorize $a^3 + 3a^2b + 3ab^2 + b^3 - 8$
Solution To expand this expression, we will use the following two Polynomial Identities :
$(x + y)^3 = x^3 + y^3 + 3xy(x + y)$ and
$(x^3 - y^3) = (x - y)(x^2 + xy + y^2)$

$\begin{align*} & a^3 + 3a^2b + 3ab^2 + b^3 - 8 \\ & = a^3 + b^3 + 3ab(a + b) - 8 \\ & = (a + b)^3 - 2^3 \\ & = (a + b - 2)((a + b)^2 + (a + b)*2 + 2^2) \\ & = (a + b - 2)((a + b)^2 + 2(a + b) + 4) \end{align*} $