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Question:

**Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients. **

**(i) ** ** **** (ii) **** (iii) ** ** **

**(iv) **** (v) ** ** (vi) **

Answer:

**(i) **** **** **

**Let P(x****) = **

**Zeroes of the polynomial are the value of ** ** where P(x) = 0.**

**Putting P(x****) = 0 then we get **

**We can find zeroes of the polynomial using splitting the middle term method.**

**In splitting the middle term method we need to find two numbers where the sum is -2 and product is -8 x**** 1 = -8.**

**Therefore, we can write,**

**Either **

**So, ** ** x = 4 and x ****= -2.**

**Let ** **= 4 and ** ** = -2 are the zeroes of the polynomial.**

**If we compare with ** ** **** then we get **

**a = 1, b = -2 and c = -8 **

**Now we verify the relationship between the zeroes and the coefficients in two different cases. **

**Case I: **

** **

**LHS = **

**RHS = ** ** **

** LHS = RHS**

**Case II: **

**LHS = ** ** = 4 x (-2)**** = - 8**

**RHS = ** ** **

** LHS = RHS**

**In both cases, LHS is equal to RHS. **

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

**(ii) **** **

**Let P(s) = **** **

**Zeroes of the polynomial are the value of ** ** where P(**s**) = 0.**

**Putting P(s****) = 0 then we get **

** **

**We can find zeroes of the polynomial using splitting the middle term method.**

**In splitting the middle term method we need to find two numbers where the sum is -4 and product is 4 ** ** 1 = 4.**

**Therefore, we can write,**

**Let **** **** and ** ** **** are the zeroes of the polynomial.**

**P(s** **) = ** ** **

**If we compare P(s) with ** ** **** then we get **

**a = 4, b = -4 and c = 1**

**Now we verify the relationship between the zeroes and the coefficients in two different cases. **

**Case I: **

** **

**LHS = **

**RHS = **

** LHS = RHS**

**Case II: **

**LHS = **

**RHS = **

** LHS = RHS**

**In both cases, LHS is equal to RHS. **

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

** (iii) **

**Let P(x** **) = **

**Zeroes of the polynomial are the value of x**** where P(x** **) = 0.**

**Putting P(x****) = 0 then we get **

**We can find zeroes of the polynomial using splitting the middle term method.**

**In splitting the middle term method we need to find two numbers where the sum is -7 and product is 6 ** ** x (-3) = -18.**

**Therefore, we can write,**

Either

**So, x**** = 3/2**** and x ****= -1/3**

**Let and ** ** are the zeroes of the polynomial.**

**P(x) = = **

**If we compare p(x) with ** ** **** then we get **

**a = 6, b = -7 and c = -3 **

**Now we verify the relationship between the zeroes and the coefficients in two different cases. **

**Case I: **

** **

**LHS = **

**RHS = **

** LHS = RHS**

**Case II: **

** **

**LHS = **

**RHS = ** ** **

** LHS = RHS**

**In both cases, LHS is equal to RHS. **

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

**(iv) **** **

**Let P(u) = **

**Zeroes of the polynomial are the value of ** ** u where P(u****) = 0.**

**Putting P(u****) = 0 then we get **

**We can find zeroes of the polynomial using common factor method.**

**4u is the common factor in the P(u) = **

**Therefore, we can write,**

**Either 4u** ** = 0 or ** ** (u+2) = 0**

**So, u**** = 0 and u ****= -2.**

**Let ** **= 0 and ** ** = -2 are the zeroes of the polynomial.**

**P(u) = **

**If we compare p(u) with **** then we get **

**a = 4, b = 8 and c = 0 **

**Now we verify the relationship between the zeroes and the coefficients in two different cases. **

**Case I: **

** **

**LHS = **

**RHS = **

** LHS = RHS**

**Case II: **

** **

**LHS = **

**RHS = **** **

** LHS = RHS**

**In both cases, LHS is equal to RHS.**

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

**(v) **** **

**Let P(t) = **** **

**Zeroes of the polynomial are the value of **** t where P(t****) = 0.**

**Putting P(t****) = 0 then we get**

So, and

**Let ** **= and ** ** = are the zeroes of the polynomial.**

**P(t) = = **

**If we compare p(t) with then we get **

**a = 1, b = 0 and c = -15**

**Now we verify the relationship between the zeroes and the coefficients in two different cases.**

**Case I: **

** **

**LHS = **

**RHS = **

** LHS = RHS**

**Case II: **

** **

**LHS = **

**RHS = **** **

** LHS = RHS**

**In both cases, LHS is equal to RHS.**

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

**(vi) **

**Let P(x****) = **

**Zeroes of the polynomial are the value of **** where P(x) = 0.**

**Putting P(x****) = 0 then we get**

**We can find zeroes of the polynomial using splitting the middle term method.**

**In splitting the middle term method we need to find two numbers where the sum is -1 and product is 3 x**** (-4) = -12.**

**Therefore, we can write,**

**Either **

**So, **** x = 4/3 and x ****= -1.**

**Let ** **= 4/3 and ** ** = -1 are the zeroes of the polynomial.**

**If we compare with **** **** then we get **

**a = 3, b = -1 and c = -4**

**Now we verify the relationship between the zeroes and the coefficients in two different cases.**

**Case I: **

** **

**LHS = **

**RHS = **** **

** LHS = RHS**

**Case II: **

**LHS = ** ** **

**RHS = ** ** **

** LHS = RHS**

**In both cases, LHS is equal to RHS.**

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

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