# When a polynomial P(x) is divided by the binomial 2x^2-3 the quotient is 2x-1 and the remainder is 3x+1. How do you find the expression of P(x)?

Dec 15, 2016

When a polynomial is divided by another polynomial, its quotient can be written as $f \left(x\right) + \frac{r \left(x\right)}{h \left(x\right)}$, where $f \left(x\right)$ is the quotient, $r \left(x\right)$ is the remainder and $h \left(x\right)$ is the divisor.

Therefore:

$P \left(x\right) = 2 x - 1 + \frac{3 x + 1}{2 {x}^{2} - 3}$

Put on a common denominator:

$P \left(x\right) = \frac{\left(\left(2 x - 1\right) \left(2 {x}^{2} - 3\right)\right) + 3 x + 1}{2 {x}^{2} - 3}$

$P \left(x\right) = \frac{4 {x}^{3} - 2 {x}^{2} - 6 x + 3 + 3 x + 1}{2 {x}^{2} - 3}$

$P \left(x\right) = \frac{4 {x}^{3} - 2 {x}^{2} - 3 x + 4}{2 {x}^{2} - 3}$

Therefore, $P \left(x\right) = 4 {x}^{3} - 2 {x}^{2} - 3 x + 4$.

Hopefully this helps!