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**Question 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

**Question 2. then what is **

⇒

⇒

⇒

**Question 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

**Question 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).

**Question 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

**Question 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)

**Question 7. What is zero of a non-zero constant polynomial? Solution: **The zero of a non-zero constant polynomial is ‘0’.

**Question 8. What is the coefficient of a zero polynomial? Solution: **The coefficient of a zero polynomial is 1.

**Question 9. What is the degree of a biquadratic polynomial? Solution: **∵ The degree of quadratic polynomial is 2.

∴ The degree of biquadratic polynomial is 4.

**Question 10. Is the statement: ‘0’ may be a zero of polynomial, true? Solution: **Yes, this statement is true.

**Question 11. What is the value of (x + a) (x + b)? Solution: **The value of (x + a) (x + b)

= x

**Question 12. What is the value of (x + y + z) ^{2 }– 2[xy + yz + zx]? Solution: **∵ (x + y + z)

= x

= x

∴ (x + y + z)

= x

**Question 13. What is the value of (x + y) ^{3} – 3xy (x + y)? Solution: **∵ (x + y)

∴ [x

⇒ (x + y)

Thus, value of x

**Question 14. Write the value of x ^{3} – y^{3}.**The value of x

Solution:

**Question 15. Write the degree of the polynomial 4x ^{4} + ox^{3} + ox^{5} + 5x + 7?**The degree of 4x

Solution:

**Question 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

**Question 17. Which of the following is one of the zero of the polynomial 2x ^{2} + 7x – 4 ? 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**

**Question 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

**Question 19. If 49x ^{2} – p =**, what is the value of p ?

**Question 20. If **** = 1, then what is the value of ** ?

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