# Polynomial Questions

## If the polynomial $$x^4 – 6x^3 + 16x^2 – 25x + 10$$ is divided by another polynomial $$x^2 – 2x + k$$, the remainder comes out to x + a, find x and a.

Solution : Let us divide $$x^4 – 6x^3 + 16x^2 – 25x + 10$$ by $$x^2 – 2x + k$$ $$\therefore$$  Remainder = (2k – 9)x – (8 – k)k + 10 But the remainder is given as x + a, On comparing their coefficients, we have : 2k – 9 = 1  $$\implies$$  k …

## If two zeroes of the polynomial $$x^4 – 6x^3 – 26x^2 + 138x – 35$$ are $$2 \pm \sqrt{3}$$, find other zeroes.

Solution : We have : $$2 \pm \sqrt{3}$$ are two zeroes of the polynomial p(x) = $$x^4 – 6x^3 – 26x^2 + 138x – 35$$ Let x = $$2 \pm \sqrt{3}$$,  So, x – 2 = $$\pm \sqrt{3}$$ Squaring, we get $$x^2 – 4x + 4$$ = 3,   i.e.  $$x^2 – 4x + 1$$ = …

## If the zeroes of the polynomial $$x^3 – 3x^2 + x + 1$$ are a – b, a and a + b, find a and b.

Solution : Since (a – b), a and (a + b) are the zeroes of the polynomials $$x^3 – 3x^2 + x + 1$$, therefore (a – b) + a + (a + b) = $$-(-3)\over 1$$ = 3 So,   3a = 3   $$\implies$$   a = 1 (a – b)a + a(a + b) + …

## Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and

Question : Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and (i)  deg p(x) = deg q(x) (ii)  deg q(x) = deg r(x) (iii)  deg q(x) = 0 Solution : (i)  Let q(x) = $$3x^2 + 2x + 6$$,                 Degree of q(x) …

## On dividing $$x^3 – 3x^2 + x + 2$$ by a polynomial g(x), the quotient and the remainder were x – 2 and -2x + 4, respectively. Find g(x).

Question : On dividing $$x^3 – 3x^2 + x + 2$$ by a polynomial g(x), the quotient and the remainder were x – 2 and -2x + 4, respectively. Find g(x). p(x) = $$x^3 – 3x^2 + x + 2$$ q(x) = x – 2 and r(x) = -2x + 4 Solution : By division …

## Obtain all the zeroes of $$3x^4 + 6x^3 – 2x^2 – 10x – 5$$, if two of its zeroes are $$\sqrt{5\over 3}$$ and -$$\sqrt{5\over 3}$$.

Solution : Since two zeroes are $$\sqrt{5\over 3}$$ and -$$\sqrt{5\over 3}$$, x = $$\sqrt{5\over 3}$$ and x = -$$\sqrt{5\over 3}$$ $$\implies$$ (x – $$\sqrt{5\over 3}$$)(x + $$\sqrt{5\over 3}$$) = $$3x^2 – 5$$ is a factor of the given polynomial. Now, we apply the division algorithm to the given polynomial and $$3x^2 – 5$$. First term …

## Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial :

Question : Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial : (i)  $$t^2 – 3$$ ; $$2t^4 + 3t^3 – 2t^2 – 9t – 12$$ (ii)  $$x^2 + 3x + 1$$ ; $$3x^4 + 5x^3 – 7x^2 + 2x + 2$$ (iii)  …

## Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each given of the following :

Question : Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each given of the following : (i)  p(x) = $$x^3 – 3x^2 + 5x – 3$$,  g(x) = $$x^2 – 2$$ (ii)  p(x) = $$x^4 – 3x^2 + 4x + 5$$, g(x) = $$x^2 + 1 – x$$ …