Question

Find all zeros of the polynomial 2x4 - 9x3 + 5x2 + 3x - 1, if two of its zeros are Rd-sharma Solutions Cbse Class 10 Mathematics Chapter - Polynomials

Answer 1

The rational root theorem can find the possible set of rational roots. The possible solutions are between $\pm \dfrac{\text{Constant term}}{\text{Leading coefficient}}$.

Now let's find the solutions based on the process above.

Let $P(x)=2x^{4}-9x^{3}+5x^{2}+3x-1$.

By rational root theorem, we have possible rational roots as $\pm\dfrac{1}{2} ,\pm1$.

At $x=1$ we have $P(1)=0$. By factor theorem, this leads to one of the factors $(x-1)$.

At $x=-\dfrac{1}{2}$ we have $P\left(-\dfrac{1}{2}\right)=\dfrac{2}{16} -\dfrac{-9}{8} +\dfrac{5}{4} +\dfrac{-3}{2} -1=\dfrac{2+18+20-24-16}{16} =0$. Similarly, this leads to one of the factors $(2x+1)$.

Now we can either choose synthetic division or long division.

Applying long division in $(2x^{4}-9x^{3}+5x^{2}+3x-1)\div(2x^{2}-x-1)$ gives quotient $x^2-4x+1$ without any remainder. Hence the factorization of $P(x)$ is $\boxed{(2x+1)(x-1)(x^{2}-4x+1)}$.

Or, we can apply synthetic division two times to find the quotient. The results of both are the same.