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# 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) x^{3} - 3x + 1, x^{5} - 4x^{3} + x^{2} + 3x + 1

**Solution:**

(i) t^{2} - 3, 2t^{4} + 3t^{3} - 2t^{2} - 9t - 12

First polynomial is given as t^{2}^{ }- 3

Second polynomial is given as 2t^{4 }+ 3t^{3}- 2t^{2 }- 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 t^{2 }- 3 is a factor of 2t^{4} + 3t^{3} - 2t^{2} - 9t - 12.

(ii) x^{2} + 3x + 1, 3x^{4} + 5x^{3} - 7x^{2} + 2x + 2

Here, first polynomial is given as x^{2 }+ 3x +1

Second polynomial is given as 3x^{4 }+ 5x^{3 }- 7x^{2 }+ 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, x^{2} + 3x + 1 is a factor of 3x^{4 }+ 5x^{3 }- 7x^{2 }+ 2x + 2.

(iii) x^{3 }- 3x + 1, x^{5 }- 4x^{3 }+ x^{2 }+ 3x + 1

Here, the first polynomial is given as x^{3} - 3x + 1

Second polynomial is given as x^{5 }- 4x^{3 }+ x^{2 }+ 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 x^{3 }- 3x + 1 is not a factor of x^{5 }- 4x^{3 }+ x^{2 }+ 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, t^{2 }− 3 and x^{2} + 3x + 1 are factors of 2t^{4} + 3t^{3} - 2t^{2} - 9t - 12 and 3x^{4} + 5x^{3} - 7x^{2} + 2x + 2 respectively since the remainder is 0 whereas x^{3} - 3x + 1 is not a factor of x^{5} - 4x^{3} + x^{2} + 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

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