If two of the zeroes of the polynomial f(x)=x4-4x3-20x2+104x-105 are 3+√2 and 3-√2,then use the division algorithm to find the other zeroes of f(x).

### Asked by saiprathumnan35 | 26th Jun, 2020, 07:17: PM

Expert Answer:

### Given polynomial is f(x)=x^{4}-4x^{3}-20x^{2}+104x-105
Two of its zeroes are 3+√2 and 3-√2
Therefore, [x - (3 + √2)] and [x - (3 - √2)] are the zeroes of f(x)
Therefore, [x - (3 + √2)][x - (3 - √2)] = [x-3-√2][x-3+√2] = x^{2} - 6x + 9 - 2 = x^{2} - 6x + 7 divides the polynomial f(x)
Using long division to divide f(x) by (x^{2} - 6x + 7), we get the quotient as x^{2} + 2x - 15
The other factor of f(x) is x^{2} + 2x - 15
x^{2} + 2x - 15 = x^{2} + 5x - 3x - 15 = x(x+5) - 3(x+5) = (x+5)(x-3)
Hence, the other two zeroes are -5 and 3.

^{4}-4x

^{3}-20x

^{2}+104x-105

^{2}- 6x + 9 - 2 = x

^{2}- 6x + 7 divides the polynomial f(x)

^{2}- 6x + 7), we get the quotient as x

^{2}+ 2x - 15

^{2}+ 2x - 15

^{2}+ 2x - 15 = x

^{2}+ 5x - 3x - 15 = x(x+5) - 3(x+5) = (x+5)(x-3)

### Answered by Renu Varma | 29th Jun, 2020, 11:21: AM

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