find all zeroes of the polynomials 2x⁴+7x³-19x²-14x+30 if two of its zeroes are √2,-√2
Two of its zeroes are √2,-√2, so two factors are, (x – √2) and (x + √2).
The product of these factors is also a factor of the original expression.
i.e. (x – √2)( x + √2) = x² - 2 divides into 2x⁴+7x³-19x²-14x+30.
Now we do a long division.
x² - 2) 2x⁴ + 7x³ - 19x² - 14x + 30 (2x² + 7x - 15
2 x⁴ - 4x²
7x³ - 15x²
7x³ - 14x
- 15x² + 30
- 15x² + 30
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So, we can now write our expression as: 2x⁴ + 7x³ - 19x² - 14x + 30 = (x² - 2)( 2x² + 7x - 15)
And the 2nd quadratic factorises as: 2x² + 7x – 15 = (2x – 3)(x + 5)
All the zeroes are: -5, 3/2, -√2, +√2