Q.1. What is p(–2) for the polynomial p(t) = t^{2} – t + 1?
Solution. p(t) = t^{2 }– t + 1
⇒ p(–2) = (–2)^{2 }– (–2) + 1
= 4 + 2 + 1
= 7
Q.2. If x  1/x = 1 then what is
Solution. On squaring both sides we get;
⇒
⇒
⇒
Q.3. If x + y = –1, then what is the value of x^{3} + y^{3} – 3xy?
Solution. We have x^{3} + y^{3} = (x + y)(x^{2} – xy + y^{2})
⇒ x^{3} + y^{3} = (–1)(x^{2 }+ 2xy + y^{2}) + 3xy
⇒ x^{3 }+ y^{3} = –1(x + y)^{2} – 3xy
⇒ x^{3 }+ y^{3} – 3xy = –1(–1)^{2} = –1(1)
⇒ x^{3} + y^{3} – 3xy = –1
Q.4. Show that p(x) is not a multiple of g(x), when p(x) = x^{3} + x – 1 g(x) = 3x – 1
Solution. g(x) = 3x – 1 = 0 ⇒ x = 1/3
∴ Remainder
Since remainder ≠ 0, so p(x) is not a multiple of g(x).
Q.5. (a) Find the value of ‘a’ if x – a is a factor of x^{3} – ax^{2} + 2x + a – 5.
(b) Find the value of ‘a’, if (x – a) is a factor of x^{3 }– ax^{2} + 2x + a – 1
[NCERT Exemplar]
(c) If x + 1 is a factor of ax^{3} + x^{2} – 2x + 4a – 9 find the value of ‘a’. [NCERT Exemplar]
Solution. (a) Let p(x) = x^{3} – ax^{2 }+ 2x + a – 5
since x – a is a factor of p(x),
so p(a) = 0
⇒ (a)^{3 }– a(a)^{2} + 2(a) + a – 5 = 0
⇒ (a)^{3} – a(a)^{2} + 2(a) + a – 5 = 0
⇒ 3a – 5 = 0 ⇒ a =
(b) Here, p(x) = x^{3} – ax^{2} + 2x + a – 1
∵ x – a is a factor of p(x)
∴ p(a) = 0
⇒ a^{3} – a(a)^{2} + 2(a) + a – 1 = 0
⇒ a^{3} – a^{3} + 2a + a – 1 = 0
⇒ 3a – 1 = 0
⇒ a = 1/3
(c) Here, x + 1 is a factor of p(x) = ax^{3 }+ x^{2} – 2x + 4a – 9
∴ p(–1) = 0
⇒ a(–1)^{3} + (–1)^{2} – 2(–1) + 4a – 9 = 0
⇒ –a + 1 + 2 + 4a – 9 = 0
⇒ 3a – 6 = 0
⇒ a = 2
Q.6. Without finding the cubes factorise (a – b)^{3} + (b – c)^{3 }+ (c – a)^{3}.
Solution. If x + y + z = 0
then x^{3} + y^{3} + z^{3 }= 3xyz
Here, (a – b) + (b – c) + (c –a) = 0
∴ (a – b)^{3 }+ (b – c)^{3} + (c – a)^{3}
= 3(a – b)(b – c)(c – a)
Q.7. What is zero of a nonzero constant polynomial?
Solution. The zero of a nonzero constant polynomial is ‘0’.
Q.8. What is the coefficient of a zero polynomial?
Solution. The coefficient of a zero polynomial is 1.
Q.9. What is the degree of a biquadratic polynomial?
Solution. ∵ The degree of a quadratic polynomial is 2.
∴ The degree of a biquadratic polynomial is 4.
Q.10. Is the statement: ‘0’ may be a zero of polynomial, true?
Solution. Yes, this statement is true.
Q.11. What is the value of (x + a) (x + b)?
Solution. The value of (x + a) (x + b)
= x^{2} + (a + b) x + ab.
Q.12. What is the value of (x + y + z)^{2 }– 2[xy + yz + zx]?
Solution. ∵ (x + y + z)^{2 }
= x^{2} + y^{2} + z^{2} + 2 xy + 2 yz + 2 zx
= x^{2} + y^{2 }+ z^{2} + 2[xy + yz + zx]
∴ (x + y + z)^{2} – 2[xy + yz + zx]
= x^{2} + y^{2} + z^{2}
Q.13. What is the value of (x + y)^{3} – 3xy (x + y)?
Solution. ∵ (x + y)^{3} = x^{3} + y^{3} + 3xy (x + y)
∴ [x^{3} + y^{3} + 3xy (x + y)] – [3xy (x + y)] = x^{3} + y^{3 }
⇒ (x + y)^{3} – 3xy(x + y) = x^{3} + y^{3}
Thus, value of x^{3 }+ y^{3} is (x + y)^{3} – 3xy (x + y)
Q.14. Write the value of x^{3} – y^{3}.
Solution. The value of x^{3 }– y^{3} is (x – y)^{3} + 3xy (x – y)
Q.15. Write the degree of the polynomial 4x^{4} + ox^{3} + ox^{5} + 5x + 7?
Solution. The degree of 4x^{4} + 0x^{3} + 0x^{5} + 5x + 7 is 4.
Q.16. What is the zero of the polynomial p(x) = 2x + 5?
Solution. ∵ p(x) = 0 ⇒ 2x + 5 = 0
⇒ x =
∴ zero of 2x + 5 is
Q.17. Which of the following is one of the zero of the polynomial 2x^{2} + 7x – 4 ?
2, 2 ?
Solution. ∵ 2x^{2} + 7x – 4 = 2x^{2} + 8x – x – 4
⇒ 2x (x + 4) – 1 (x + 4) = 0
⇒ (x + 4) (2x – 1) = 0
⇒ x = – 4, x = 1/2,
∴ One of the zero of 2x^{2} + 7x – 4 is 1/2.
Q.18. If a + b + 2 = 0, then what is the value of a^{3} + b^{3} + 8.
Solution. ∵ x + y + z = 0
⇒ x^{3 }+ y^{3} + z^{3} = 3xyz
∴ a + b + 2 = 0
⇒ (a)^{3 }+ (b)^{3} + (2)^{3} = 3(a × b × 2) = 6ab
⇒ The value of a^{3} + b^{3 }+ 8 is 6ab
Q.19. If 49x^{2} – p =, what is the value of p?
Solution.
Q.20. If = 1, then what is the value of ?
Solution. On squaring both the sides we get;
48 videos378 docs65 tests

1. What is a polynomial? 
2. How to determine the degree of a polynomial? 
3. Can a polynomial have negative exponents? 
4. What is the difference between a monomial and a polynomial? 
5. How to perform polynomial division? 
48 videos378 docs65 tests


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