Find the zeroes of the polynomial 2+2√2x -6 and verify the relationship between its zeroes and coefficients
Find the zeroes of the polynomial
2+2√2x -6 and verify the relationship between its zeroes and
coefficients
ANSWER:
To find the zeroes of the polynomial 2 + 2√2x - 6, we set the polynomial equal to zero and solve for x:
2 + 2√2x - 6 = 0
Rearranging the terms:
2√2x = 4
Dividing both sides by 2√2:
x = 2/√2
To simplify the expression, we rationalize the denominator:
x = (2/√2) * (√2/√2) = 2√2/2 = √2
Therefore, the zero of the polynomial is x = √2.
To verify the relationship between the zeroes and coefficients of the polynomial, we can use Vieta's formulas. For a quadratic polynomial in the form ax^2 + bx + c = 0, the sum of the zeroes is given by -b/a and the product of the zeroes is given by c/a.
In this case, the polynomial 2 + 2√2x - 6 is not a quadratic polynomial, but rather a linear one. However, we can still apply Vieta's formulas:
The sum of the zeroes is -b/a = -(2√2)/1 = -2√2.
The product of the zeroes is c/a = (-6)/1 = -6.
Therefore, the relationship between the zeroes (√2) and the coefficients (2, 2√2, -6) is that the sum of the zeroes (-2√2) is equal to the negation of the coefficient of the linear term, and the product of the zeroes (-6) is equal to the constant term of the polynomial.
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