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Polynomials CBSE NCERT Solutions
Obtain all other zeroes of 3x4 + 6x3 - 2x2 - 10x - 5, if two of its zeroes are and .
Solutions: Suppose P(x) = 3x4 + 6x3 - 2x2 - 10x – 5.
Since 3x4 + 6x3 - 2x2 - 10x – 5 is a polynomial of degree 4, so there will be a total of 4 roots.
and are zeroes of polynomial P(x).
(3x2 – 5 ) = 0 is a factor of given polynomial P(x).
Now, when we will divide P(x) by (3x2 − 5) the quotient obtained will also be a factor of P(x) and the remainder will be 0.
Therefore, 3x4 + 6x3 −2x2 −10x – 5 = (3x2 – 5)(x2 + 2x + 1)
We can find zeroes of the polynomial (x2 + 2x + 1) using splitting the middle term method.
In splitting the middle term method we need to find two numbers where the sum is 1 and product is 1 x 1 = 2.
Therefore, we can write,
x2 + x + x + 1 = 0
x ( x + 1) + 1( x + 1) = 0
(x + 1)(x + 1) = 0
So, x = −1 and x = −1.
Hence, all four zeroes of polynomial P(x) = 3x4 + 6x3 - 2x2 - 10x – 5 are , , −1 and −1.
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