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

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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