Question 30 Obtain all the zeros of the polynomial x4 + 4x3 − 2x2 − 20x − 15, if two of its zeroes are √5 and −√5.
Let p(x) = x4 + 4x3 − 2x2 − 20x − 15
Since x = √5 is a zero , x – √5 is a factor
Since x = – √5 is a zero , x + √5 is a factor
Hence ,
(x + √5) (x – √5) is a factor
i.e. (x2 – (√5)^2) is also a factor
i.e. (x2 – 5) is also a factor
Now, by dividing the given polynomial by (x2 – 5)
We can find out other factors
Now,
we factorize x2 + 4x + 3
x2 + 4x + 3
We use splitting the
middle term method
= x2 + 3x + x + 3
= x(x + 3) + 1(x + 3)
= (x + 1)(x + 3)
Splitting the middle term method
We need to find two numbers whose
Sum = 4
Product = 3 × 1 = 3
∴ x = –1 & x = –3 are the zeroes of p(x)
Therefore,
the zeroes of p(x) are √5, –√5, –1, and –3

Made by

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.