Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:
(i) t2 - 3, 2t4 + 3t3 - 2t2 - 9t - 12
(ii) x2 + 3x + 1, 3x4 + 5x3 - 7x2 + 2x + 2
(iii) x3 - 3x + 1, x5 - 4x3 + x2 + 3x + 1
Solution:
(i) t2 - 3, 2t4 + 3t3 - 2t2 - 9t - 12
First polynomial is given as t2 - 3
Second polynomial is given as 2t4 + 3t3- 2t2 - 9t -12
To check whether the first polynomial is a factor of the second polynomial, the remainder must be equal to zero.
Let us divide and observe the remainder.
As we can see, that the remainder is equal to 0.
Therefore, we say that the given polynomial t2 - 3 is a factor of 2t4 + 3t3 - 2t2 - 9t - 12.
(ii) x2 + 3x + 1, 3x4 + 5x3 - 7x2 + 2x + 2
Here, first polynomial is given as x2 + 3x +1
Second polynomial is given as 3x4 + 5x3 - 7x2 + 2x + 2
To check whether the given polynomial is a factor, the remainder must be equal to zero.
Let us divide and observe the remainder.
As we can see, the remainder is left as 0. Therefore, we can say that, x2 + 3x + 1 is a factor of 3x4 + 5x3 - 7x2 + 2x + 2.
(iii) x3 - 3x + 1, x5 - 4x3 + x2 + 3x + 1
Here, the first polynomial is given as x3 - 3x + 1
Second polynomial is given as x5 - 4x3 + x2 + 3x + 1
To check whether the given polynomial is a factor, the remainder must be equal to zero.
Let us divide and observe the remainder.
As we can see, the remainder is not equal to 0. Therefore, we can say that x3 - 3x + 1 is not a factor of x5 - 4x3 + x2 + 3x + 1.
☛ Check: NCERT Solutions for Class 10 Maths Chapter 2
Video Solution:
Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial: (i) t² - 3, 2t⁴ + 3t³ - 2t² - 9t - 12 (ii) x² + 3x + 1, 3x⁴ + 5x³ - 7x² + 2x + 2 (iii) x³ - 3x + 1, x⁵ - 4x³ + x² + 3x + 1
NCERT Solutions Class 10 Maths Chapter 2 Exercise 2.3 Question 2
Summary:
By checking whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial we see that, t2 − 3 and x2 + 3x + 1 are factors of 2t4 + 3t3 - 2t2 - 9t - 12 and 3x4 + 5x3 - 7x2 + 2x + 2 respectively since the remainder is 0 whereas x3 - 3x + 1 is not a factor of x5 - 4x3 + x2 + 3x + 1 since the remainder is not 0.
☛ Related Questions:
- Obtain all other zeroes of 3x4 + 6x3 - 2x2 - 10x - 5, if two of its zeroes are √ 5/3 and - √ 5/3
- On dividing x3 - 3x2 + x + 2 by a polynomial g(x), the quotient and remainder were x - 2 and - 2x + 4, respectively. Find g (x).
- 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
- Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following:(i) p(x) = x3 - 3x2 + 5x - 3, g(x) = x2 - 2(ii) p(x) = x4 - 3x2 + 4x + 5, g(x) = x2 + 1 - x(iii) p(x) = x4 - 5x + 6, g(x) = 2 - x2