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info@snapandsolve.comPolynomial Function:
To write a polynomial function with the given zeros 2 and 3 + √7, we must also include the conjugate of 3 + √7, which is 3 - √7, to ensure the coefficients are rational.
The polynomial is formed by multiplying the factors corresponding to each zero:
Multiply these factors:
(x - 2) [(x - 3) - √7][(x - 3) + √7]
Using the difference of squares:
(x - 2) [(x - 3)² - (√7)²]
(x - 2) [(x - 3)² - 7]
Expand (x - 3)²:
(x - 2) [x² - 6x + 9 - 7]
(x - 2) [x² - 6x + 2]
Expand the product:
x(x² - 6x + 2) - 2(x² - 6x + 2)
x³ - 6x² + 2x - 2x² + 12x - 4
Combine like terms:
x³ - 8x² + 14x - 4
Thus, the polynomial function is:
f(x) = x³ - 8x² + 14x - 4
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