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Question:

**Obtain all other zeroes of 3x ^{4 }+ 6x^{3 }- 2x^{2 }- 10x - 5, if two of its zeroes are **

**and**

**.**

Answer:

**Solutions: Suppose P(x) = 3x ^{4 }+ 6x^{3 }- 2x^{2 }- 10x – 5. **

**Since 3x ^{4 }+ 6x^{3 }- 2x^{2 }- 10x – 5 is a polynomial of degree 4, so there will be a total of 4 roots.**

**and **** ** **are zeroes of polynomial P(x).**

** **** ** ** **

**(3x ^{2 }– 5 ) = 0 is a factor of given polynomial P(x).**

**Now, when we will divide P(x) by (3x ^{2 }− 5) the quotient obtained will also be a factor of P(x) and the remainder will be 0.**

**Therefore, 3x ^{4 }+ 6x^{3 }−2x^{2 }−10x – 5 = (3x^{2 }– 5)(x^{2 }+ 2x + 1)**

**We can find zeroes of the polynomial ****(x ^{2 }+ 2x + 1) **

**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 1 ** **x 1 = 2.**

**Therefore, we can write,**

** x ^{2 }+ x + x + 1 = 0**

** x ( x + 1) + 1( x + 1) = 0**

** (x + 1)(x + 1) = 0**

**So, x = −1 and x = −1.**

**Hence, all four zeroes of polynomial P(x) = 3x ^{4 }+ 6x^{3 }- 2x^{2 }- 10x – 5 are**

**,**

**, −1 and −1.**

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