Worksheet: Polynomials Notes | Study Mathematics (Maths) Class 10 - Class 10

1. Find the quadratic polynomial whose zeroes arerespectively.
2. Find the zeroes of the polynomial – 2.
3. Divide: x³ − 1 by x + 1.
4. Find the quadratic polynomial whose zeroes are  respectively.
5. Divide: x⁴ − 1 by x + 1.
6. Divide: 4x³ − x + 1 by 2x − 1.
7. Divide: x⁴ + x² + 1 by x² + x + 1.
8. Find the quotient and remainder when x⁴ + 1 is divided by x − 1.
9. Find the quotient and remainder when f (x) = 4x³ − 12x² + 14x − 13 is divided by
10. Find the quotient and remainder when 3x⁴ + 5x³ –7x² + 2x + 2 is divided by x² + 3x + 1.
11. Find the remainder when x⁴ – 3x² + 4x + 5 is divided by x² – x + 1.
12. 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).
13. Find the zeroes of 2x⁴ − 3x³ − 3x² + 6x − 2, if you know that two of its zeroes are √2 and -√2.
14. If α and β are the zeroes of the quadratic polynomial f (x) = x² − 3x + 2, then find the value of α² + β².
15. It is given that 1 is one of the zeroes of the polynomial 7x − x³ − 6. Find its other zero.
Hint: (− x³ + 7x − 6) ÷ (x − 1)= − (x² + x − 6) = − [(x + 3) (x - 2)]
∴ The other zeroes are −3 and 2.
16. Divide 2x⁴ − 9x³ + 5x² + 3x − 8 by x² − 4x + 1 and verify the division by algorithm.
Hint: (2x⁴ − 9x³ + 5x² + 3x − 8) ÷ (x² − 4x + 1)
⇒ quotient = 2x² − x − 1 and remainder = −7
Verification: 
(Quotient × Divisor) + Remainder = Dividend or (2x² − x − 1) × (x² − 4x + 1) + (− 7) = 2x⁴ − 9x³ + 5x² + 3x − 8
17. Find k so that x² + 2x + k is a factor of 2x⁴+ x³ – 14x² + 5x + 6. Also, find all the zeroes of the polynomial.
Hint: For (x² + 2x + k) to be a factor the remainder in (2x⁴ + x³ – 14x² + 5x + 6) ÷ (x² + 2x + k) must be zero.
18. If α, β are the zeroes of the quadraticpolynomial p(x) = x² – (k–6)x + (2k + 1), find the value of k when α + β = αβ.
Hint: Sum of roots α + β = –[–(k – 6)]= k – 6
Product of roots αβ = 2k + 1
∵ α + β = αβ
∴ k – 6 = 2k + 1
⇒ (2k– k) = – 6 – 1
∴ k = –7
19. Find the zeroes of the polynomial p(x) =  and verify the relationship between the zeroes and the coefficients.
Hint:

In the given polynomial,


Product of roots
From (1), (2) and (3), the relation between the zeroes and coefficient is verified.

ANSWERS
1. x² – 10x + 7
2.
3. x² – x + 1; Remainder = –2
4. x² – 10x + 21
5. x³ + x² + x + 1; Remainder = – 2
6. 2x² + x; Remainder = 1
7. x² – x + 1; Remainder = 0
8. x³ + x² + x + 1; Remainder = 2
9. 4x² – 10x + 19, Remainder = -7/2
10. 3x² – 4x + 2; Remainder = 0
11. Quotient = x² + x –3; Remainder = 8
12. g(x) = x² – x + 1
13.
14. 5
15. –3 and 2
16. 2x² – x – 1; Remainder = – 7
17. k = – 3; zeroes;
18. k = –7,
19.