Question

Factorise x³ - 10 x²-x + 10​

Answer 1

Answer:

x³ - 10x² - x + 10 = (x - 1) (x - 10) (x + 1)

Step-by-step explanation:

Given cubic polynomial :

x³ - 10x² - x + 10

Method - 1 :

x³ - 10x² - x + 10

x² (x - 10) - 1(x - 10)

(x - 10) (x² - 1)

(x - 10) (x² - 1²)  

(x - 10) (x - 1) (x + 1)   [ ∵ a² - b² = (a - b) (a + b) ]

Method - 2 :

Note : If the sum of all the coefficients of the cubic polynomial is equal to zero, then (x - 1) is a factor of the given cubic polynomial.

Let's see if (x - 1) is a factor of the given cubic polynomial.

⇒ Sum of the coefficients

= 1 + (-10) + (-1) + 10

= 1 - 10 - 1 + 10

= 0

The result is zero. Hence (x - 1) is a factor of the given cubic polynomial.

i.e., 1 is a zero of the given polynomial.

Now, divide the given polynomial x³ - 10x² - x + 10 by (x - 1)

       

Quotient = x² - 9x - 10

Now, we have to factorize x² - 9x - 10

x² - 9x - 10

x² - 10x + x - 10

x(x - 10) + 1(x - 10)

(x - 10) (x + 1)

x² - 9x - 10 = (x - 10) (x + 1)

Therefore, x³ - 10x² - x + 10 = (x - 1) (x - 10) (x + 1)

Answer 2

x3 – 10x2 – x + 10 Let p(x) = x3 – 10x2 – x + 10 Sum of the co-efficients = 1 – 0 – 1 + 10 = 11 – 11 = 0 ∴ (x – 1) is a factor Sum of co-efficients of even powers of x with constant = -10 + 10 = 0 Sum of co-efficients of odd powers of = 1 – 1 = 0 ∴ (x + 1) is a factor Synthetic division

∴ x3 + 10x2 – x + 10 = (x – 1) (x + 1) (x – 10)