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

Answer:

**Solutions:**

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

**We have,**

**The first polynomial is t ^{2 }- 3 and the second polynomial is 2t^{4 }+ 3t^{3 }- 2t^{2 }- 9t -12. **

**Here, remainder is zero. **

**Hence, t ^{2 }-3 is a factor of 2t^{2 }+ 3t+ 4.**

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

**We have,**

**First polynomial is x ^{2 }+ 3x + 1 and Second polynomial is 3x^{4 }+ 5x^{3 }- 7x^{2 }+ 2x + 2.**

**Here, the remainder is left as 0. **

**Hence, 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**

**We have,**

**First polynomial is x ^{3 }- 3x + 1 and Second polynomial is x^{5}- 4x^{3 }+ x^{2 }+ 3x + 1.**

**Hence, the remainder is not equal to 0. **

**Hence, x ^{3}- 3x + 1 is not a factor of x^{5}- 4x^{3 }+ x^{2 }+ 3x + 1 .**

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