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**Question 1. What is p(â€“2) for the polynomial p(t) = t ^{2} â€“ t + 1?**

**Solution:** p(t) = t^{2 }â€“ t + 1

â‡’ p(â€“2) = (â€“2)^{2 }â€“ (â€“2) + 1

= 4 + 2 + 1

= 7

**Question 2. then what is **

â‡’

â‡’

â‡’

**Question 3. If x + y = â€“1, then what is the value of x ^{3} + y^{3} â€“ 3xy? **

**Solution: **We have x^{3} + y^{3} = (x + y)(x^{2} â€“ xy + y^{2})

â‡’ x^{3} + y^{3} = (â€“1)(x^{2 }+ 2xy + y^{2}) + 3xy

â‡’ x^{3 }+ y^{3} = â€“1(x + y)^{2} â€“ 3xy

â‡’ x^{3 }+ y^{3} â€“ 3xy = â€“1(â€“1)^{2} = â€“1(1)

â‡’ x^{3} + y^{3} â€“ 3xy = â€“1

**Question 4. Show that p(x) is not a multiple of g(x), when p(x) = x ^{3} + x â€“ 1 g(x) = 3x â€“ 1**

**Solution: **g(x) = 3x â€“ 1 = 0 â‡’ x = 1/3

âˆ´ Remainder

Since remainder â‰ 0, so p(x) is not a multiple of g(x).

**Question 5. (a) Find the value of â€˜aâ€™ if x â€“ a is a factor of x ^{3} â€“ ax^{2} + 2x + a â€“ 5.**

(b) Find the value of â€˜aâ€™, if (x â€“ a) is a factor of x^{3 }â€“ ax^{2} + 2x + a â€“ 1 [NCERT Exemplar]

(c) If x + 1 is a factor of ax^{3} + x^{2} â€“ 2x + 4a â€“ 9 find the value of â€˜aâ€™. [NCERT Exemplar]

**Solution: **(a) Let p(x) = x^{3} â€“ ax^{2 }+ 2x + a â€“ 5

since x â€“ a is a factor of p(x),

so p(a) = 0

â‡’ (a)^{3 }â€“ a(a)^{2} + 2(a) + a â€“ 5 = 0

â‡’ (a)^{3} â€“ a(a)^{2} + 2(a) + a â€“ 5 = 0

â‡’ 3a â€“ 5 = 0 â‡’ a =

(b) Here, p(x) = x^{3} â€“ ax^{2} + 2x + a â€“ 1

âˆµ x â€“ a is a factor of p(x)

âˆ´ p(a) = 0

â‡’ a^{3} â€“ a(a)^{2} + 2(a) + a â€“ 1 = 0

â‡’ a^{3} â€“ a^{3} + 2a + a â€“ 1 = 0

â‡’ 3a â€“ 1 = 0

â‡’ a = 1/3

(c) Here, x + 1 is a factor of p(x) = ax^{3 }+ x^{2} â€“ 2x + 4a â€“ 9

âˆ´ p(â€“1) = 0

â‡’ a(â€“1)^{3} + (â€“1)^{2} â€“ 2(â€“1) + 4a â€“ 9 = 0

â‡’ â€“a + 1 + 2 + 4a â€“ 9 = 0

â‡’ 3a â€“ 6 = 0

â‡’ a = 2

**Question 6. Without finding the cubes factorise (a â€“ b) ^{3} + (b â€“ c)^{3 }+ (c â€“ a)^{3}.**

**Solution: **If x + y + z = 0

then x^{3} + y^{3} + z^{3 }= 3xyz

Here, (a â€“ b) + (b â€“ c) + (c â€“a) = 0

âˆ´ (a â€“ b)^{3 }+ (b â€“ c)^{3} + (c â€“ a)^{3}

= 3(a â€“ b)(b â€“ c)(c â€“ a)

**Question 7. What is zero of a non-zero constant polynomial? Solution: **The zero of a non-zero constant polynomial is â€˜0â€™.

**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: **âˆµ The degree of quadratic polynomial is 2.

âˆ´ The degree of biquadratic polynomial is 4.

**Question 10. Is the statement: â€˜0â€™ 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 }â€“ 2[xy + yz + zx]? Solution: **âˆµ (x + y + z)

= x

= x

âˆ´ (x + y + z)

= x

**Question 13. What is the value of (x + y) ^{3} â€“ 3xy (x + y)? Solution: **âˆµ (x + y)

âˆ´ [x

â‡’ (x + y)

Thus, value of x

**Question 14. Write the value of x ^{3} â€“ 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:** âˆµ p(x) = 0 â‡’ 2x + 5 = 0

â‡’ x =

âˆ´ zero of 2x + 5 is

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

**Solution**: âˆµ 2x^{2} + 7x â€“ 4 = 2x^{2} + 8x â€“ x â€“ 4

â‡’ 2x (x + 4) â€“ 1 (x + 4) = 0

â‡’ (x + 4) (2x â€“ 1) = 0

â‡’ x = â€“ 4, x = **1/2,**

âˆ´ One of the zero of 2x^{2} + 7x â€“ 4 is **1/2**

**Question 18. If a + b + 2 = 0, then what is the value of a ^{3} + b^{3} + 8.**

**Solution:** âˆµ x + y + z = 0

â‡’ x^{3 }+ y^{3} + z^{3} = 3xyz

âˆ´ a + b + 2 = 0

â‡’ (a)^{3 }+ (b)^{3} + (2)^{3} = 3(a Ã— b Ã— 2) = 6ab

â‡’ The value of a^{3} + b^{3 }+ 8 is 6ab

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

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

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