| --- |
| source: ncert_exercises |
| topic: algebra |
| class_level: class_10 |
| chapter: ch2 |
| exercise: ex2.3 |
| difficulty: intermediate |
| --- |
| |
| Class 10 | Ch2: Polynomials | Exercise 2.3 |
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| Division Algorithm for Polynomials: |
| Dividend = Divisor × Quotient + Remainder |
| p(x) = g(x) × q(x) + r(x) |
| where degree(r) < degree(g) or r=0. |
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| Q1. Divide the polynomial p(x) by the polynomial g(x) and find quotient and remainder: |
| (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 |
| Answer: |
| (i) x³-3x²+5x-3 ÷ (x²-2): Quotient=x-3, Remainder=7x-9. ✅ |
| (ii) x⁴-3x²+4x+5 ÷ (x²-x+1): Quotient=x²+x-3, Remainder=8. ✅ |
| (iii) x⁴-5x+6 ÷ (2-x²)=x⁴-5x+6 ÷ (-x²+2): Quotient=-x²-2, Remainder=-5x+10. ✅ |
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| Q2. Check whether first polynomial is a factor of second: |
| (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 |
| Answer: |
| (i) Divide: Remainder=0. Yes, t²-3 IS a factor. ✅ |
| (ii) Divide: Remainder=0. Yes, IS a factor. ✅ |
| (iii) Divide: Remainder=2. NOT a factor. ✅ |
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| Q3. Obtain all zeroes of 3x⁴+6x³-2x²-10x-5 if two zeroes are √(5/3) and -√(5/3). |
| Answer: Two zeroes = ±√(5/3), so (x-√(5/3))(x+√(5/3))=x²-5/3 is a factor. |
| Divide: 3x⁴+6x³-2x²-10x-5 ÷ (3x²-5) = x²+2x+1 = (x+1)². Remaining zeroes: -1,-1. All zeroes: √(5/3), -√(5/3), -1, -1. ✅ |
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| Q4. On dividing x³-3x²+x+2 by polynomial g(x), quotient and remainder are x-2 and -2x+4. Find g(x). |
| Answer: p(x)=g(x)×q(x)+r(x). g(x)=[p(x)-r(x)]/q(x)=(x³-3x²+x+2-(-2x+4))/(x-2)=(x³-3x²+3x-2)/(x-2)=x²-x+1. ✅ |
