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Question:

**Verify that the numbers given along side of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:**

**2x**^{3}+ x^{2}- 5x + 2; 1/2**, 1, -2**-
**x**^{3}- 4x^{2}+ 5x – 2; 2, 1, 1

Answer:

**(i) 2x ^{3} + x^{2} - 5x + 2; 1/2**

**, 1, -2**

**Suppose p(x) = 2x ^{3} + x^{2} - 5x + 2 ……….. (I)**

**Now we verify ****, 1 and -2 are zeroes of the polynomial**

**On substituting the value of x = ****โ in equation (I), we get**

**2(โ )**^{3}** + (** **โ ) ^{2} – 5(**

**โ )**

**+ 2**

**= -2 + 2 = 0**

**On substituting the value of x = 1 in equation (I), we get**

**2(1)**^{3}** + (**1**)**^{2}** – 5(**1**)**** + 2**

**= 2 + 1 – 5 + 2**

**= 5 - 5 = 0**

**On substituting the value of x = -2 in equation (I), we get**

**2(-2)**^{3}** + (-2****)**^{2}** – 5(-2****)**** + 2**

**= 2(-8) + 4 + 10 + 2**

**= -16 + 16 = 0**

**Therefore, ** **, 1 and -2**** are the zeroes of the given polynomial.**

**Now we can Compare the polynomial ****2x ^{3} + x^{2} - 5x + 2 **

**with**

**ax**

^{3}+ bx^{2}+ cx + d,**we obtain,**

**a ****= 2,b = 1,c = −5, d = 2**

**Let us assume α =** **โ โ, β = 1, ** **= −2**

**Sum of the roots = α + β + ** **= โ + 1 – 2 = โ** **โ – 1= = **

**αβ + β** ** + ****α** ** = ****โ** ** **

** **

**Product of the roots = αβ** ** **** โ** ** **

**Therefore, the relationship between the zeroes and the coefficient are verified.**

**(ii) x ^{3} - 4x^{2} + 5x – 2; 2, 1, 1**

**Suppose p(x) = x ^{3} - 4x^{2} + 5x – 2 ……….. (I)**

**Now we verify 2****, 1 and 1 are zeroes of the polynomial.**

**On substituting the value of x = 2โ in equation (I), we get**

** 2 ^{3} – 4(2)^{2} + 5(2) – 2 **

** 8 – 16 + 10 – 2**

**= 18 – 18 = 0**

**On substituting the value of x = 1 in equation (I), we get**

** 1 ^{3} – 4(1)^{2} + 5(1) – 2 **

**= 1 - 4 + 5 - 2**

**= 6 - 6 = 0**

**Therefore, 2****, 1 and 1**** are the zeroes of the given polynomial.**

**Now we can Compare the polynomial ****x ^{3} - 4x^{2} + 5x – 2 **

**with**

**ax**

^{3}+ bx^{2}+ cx + d,**we obtain,**

**a ****= 1,b = -4, c = 5, d = -2**

**Let us assume α =** **โ, β = 1, ** **= 1**

**Sum of the roots = α + β + ** **= 2 + 1 + 1 = โ4**** = **

**αβ + β** ** + ****α** ** = 2โx****1 + 1x1** **+ 2x1 โ** ** = 2 + 1 + 2 = 5 ****= **

**Product of the roots = αβ** ** = 1x1x1 โ** ** = 1 = **

**Therefore, the relationship between the zeroes and the coefficient are verified.**

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