Q1. The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case
Solution 1
Q1. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.
(i) `x^2` – 2x – 8 (ii) 4`s^2` – 4s + 1 (iii) 6`x^2` – 3 – 7x (iv) 4`u^2` + 8u (v) `t^2` – 15 (vi) 3`x^2` – x – 4
(i) `x^2` – 2x – 8 (ii) 4`s^2` – 4s + 1 (iii) 6`x^2` – 3 – 7x (iv) 4`u^2` + 8u (v) `t^2` – 15 (vi) 3`x^2` – x – 4
Solution 1
Q2. Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.
(i) `frac{1}{4}` ,− 1 (ii) `sqrt{2}` , `frac{1}{3}` (iii) 0, `sqrt{5}` (iv) 1, 1 (v) -`frac{1}{4}`,`frac{1}{4}` (vi) 4, 1
(i) `frac{1}{4}` ,− 1 (ii) `sqrt{2}` , `frac{1}{3}` (iii) 0, `sqrt{5}` (iv) 1, 1 (v) -`frac{1}{4}`,`frac{1}{4}` (vi) 4, 1
Solution 1
Q1. Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following:
(i) p(x) = `x^3` – 3`x^2` + 5x – 3, g(x) = `x^2` – 2
(ii) p(x) = `x^4` – 3x 2 + 4x + 5, g(x) = `x^2` + 1 – x
(iii) p(x) = `x^4` – 5x + 6, g(x) = 2 – `x^2`
(i) p(x) = `x^3` – 3`x^2` + 5x – 3, g(x) = `x^2` – 2
(ii) p(x) = `x^4` – 3x 2 + 4x + 5, g(x) = `x^2` + 1 – x
(iii) p(x) = `x^4` – 5x + 6, g(x) = 2 – `x^2`
Solution 1
Q2. 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, 2`t^4` + 3`t^3` – 2`t^2` – 9t – 12
(ii) `x^2` + 3x + 1, 3`x^4` + 5`x^3` – 7`x^2` + 2x + 2
(iii) `x^3` – 3x + 1, `x^5` – 4x 3 + `x^2` + 3x + 1
(i) `t^2`– 3, 2`t^4` + 3`t^3` – 2`t^2` – 9t – 12
(ii) `x^2` + 3x + 1, 3`x^4` + 5`x^3` – 7`x^2` + 2x + 2
(iii) `x^3` – 3x + 1, `x^5` – 4x 3 + `x^2` + 3x + 1
Solution 1
Q3. Obtain all other zeroes of 3`x^4` + 6`x^3` – 2`x^2` – 10x – 5, if two of its zeroes are `sqrtfrac{5}{3}` and –`sqrtfrac{5}{3}` ⋅
Solution 1
Q4. 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).
Solution 1
Q5. 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 r(x) = 0
(i) deg p(x) = deg q(x) (ii) deg q(x) = deg r(x) (iii) deg r(x) = 0
Solution 1
Q1. Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:
(i) 2`x^3` + `x^2` – 5x + 2; `frac{1}{2}, 1, – 2 (ii) `x^3` – 4`x^2` + 5x – 2; 2, 1, 1
(i) 2`x^3` + `x^2` – 5x + 2; `frac{1}{2}, 1, – 2 (ii) `x^3` – 4`x^2` + 5x – 2; 2, 1, 1
Solution 1
Q2. Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, –7, –14 respectively.
Solution 1
Q3. If the zeroes of the polynomial `x^3` – 3`x^2` + x + 1 are a – b, a, a + b, find a and b.
Solution 1
Q4. If two zeroes of the polynomial `x^4` – 6`x^3` – 26`x^2` + 138x – 35 are 2 ± `sqrt{3}` find other zeroes.
Solution 1
Q5. If the polynomial `x^4` – 6`x^3` + 16`x^2` – 25x + 10 is divided by another polynomial `x^2` – 2x + k, the remainder comes out to be x + a, find k and a.
Solution 1