# Ex.2.3 Q4 Polynomials Solution - NCERT Maths Class 10

## Question

On dividing \(x^{3} -3 x^{2}+x +2\) by a

polynomial \(g(x),\) the quotient and remainder were \(x – 2\) and \(–2x + 4,\) respectively. Find \(g(x).\)

## Text Solution

**What is unknown?**

Divisor *\(g(x)\)* of a polynomial \(p(x).\)

**Reasoning:**

This question is straight forward, you can solve it by using division algorithm

Dividend \(=\) Divisor \(\times\) Quotient \(+\) Remainder

Put the given values in the above equation and simplify it, get the value of \(g(x).\)

**Steps:**

Dividend \(=\) Divisor \(\times\) Quotient \(+\) Remainder

\(\begin{align}&x^{3}\!-\!3 x^{2}\!+\!x\!+\!2\!=\!g(x) \!\times\! x\!-\!2\!+\!(-2 x\!+\!4)\\&\!\left(x^{3}\!-\!3 x^{2}\!+\!x\!+\!2\right)\!\!-\!\!(\!\!-\!2 x\!+\!4)\!=\!g(x)\! \times\! x\!-\!2\\&\left(x^{3}\!-\!3 x^{2}\!+\!x\!+\!2 x\!+\!2\!-\!4\right)\!=\!g(x)\! \times\! x\!-\!2\\&\left(x^{3}\!-\!3 x^{2}\!+\!3 x\!-\!2\right)\!=\!g(x) \times x\!-\!2 \end{align}\)

Therefore, \(g\left( x \right) = {x^2} - x + 1\)