Factorising the cubic polynomial — lesson. Mathematics CBSE, Class 9.

Steps to factorize the cubic polynomial \(p(x)\).

Step 1: Find \(x = a\) where \(p(a) = 0\). That is, we have to find one of the factors.

Step 2: Then \(x-a\) is a factor of \(p(x)\).

Step 3: Now divide \(p(x)\) by \(x-a\). That is \(\frac{p(x)}{(x-a)}\).

Step 4: Then factorize the quotient(quadratic equation) by splitting its middle term.

Step 1: Let us find one of factors by trial method.

Consider the polynomial

m^{3} - m^{2} + 13 m \underline{- 13}

The product of coefficient of \(m^3\) and constant \(=\) \(1 \times -13\) \(=\) \(-13\).

We shall now look for the factors of \(-13\).

Factors of \(-13\): \(\pm1, \pm13\)

Let us start with the first factor \(m = 1\).

\begin{array}{l} p (1) = {(1)}^{3} - {(1)}^{1} + 13 (1) - 13 \\ p (1) = (1) - (1) + 13 - 13 \\ p (1) = 1 - (1) + 13 - 13 \\ P (1) = 0 \\ P (1) = 0 + 0 = 0 \end{array}

So at \(m =1\),

p (y) = 0

Thus, the factor \(m =1\) satisfies step 1.

Step 2:

We can conclude that \(m -1\) is a factor of

p (k)

Step 3: Now divide

p (k)

by \(m -1\).

Let the quotient be \(g(m)\).

\begin{array}{l} g (m) = \frac{p (m)}{(m - 1)} \\ m^{2} + 13 \\ m - 1 m^{3} - m^{2} + 13 m - 13 \\ m^{3} - m^{2} \\ \underline{(-) (+)} \\ 0 + 13 m - 13 \\ 13 m - 13 \\ \underline{(-) (+)} \\ 0 \end{array}

So, \(g(m) =\)

m^{2} + 13

\begin{array}{l} p (m) = (m - 1) g (m) \\ = (m - 1) (m^{2} + 13) \end{array}

Thus, the factorisation of the cubic polynomial

m^{3} - m^{2} + 13 m - 13

(m - 1) (m^{2} + 13)

Did you find an error?

3. Factorising the cubic polynomial