Polynomial Questions

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 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. Read More »

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 two zeroes of the polynomial \(x^4 – 6x^3 – 26x^2 + 138x – 35\) are \(2 \pm \sqrt{3}\), find other zeroes. Read More »

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) + …

If the zeroes of the polynomial \(x^3 – 3x^2 + x + 1\) are a – b, a and a + b, find a and b. Read More »

Solution : Let the cubic polynomial be \(ax^3 + bx^2 + cx + d\), and its zeroes be \(\alpha\), \(\beta\) and \(\gamma\). Then,  \(\alpha\) + \(\beta\) + \(\gamma\) = 2 = \(-(-2)\over 1\) = \(-b\over a\) \(\alpha\)\(\beta\) + \(\beta\)\(\gamma\) + \(\gamma\)\(\alpha\) = – 7 = \(-7\over 1\) = \(c\over a\) and \(\alpha\)\(\beta\)\(\gamma\) = -14 = \(-14\over …

Find the cubic polynomial with the sum, sum of the products of its zeroes taken two at a time, and the product of its zeroes as 2, – 7, -14 respectively. Read More »

Question : Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficient in each case : (i)  \(2x^3 + x^2 – 5x + 2\);  \(1\over 2\), 1, -2 (ii)  \(x^3 – 4x^2 + 5x – 2\);  2, 1, 1 Solution : …

Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficient in each case : Read More »

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) …

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

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 …

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). Read More »

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 …

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

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)  …

Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial : Read More »

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) = \(x^2 + 1 – x\) …

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