Find the zeroes of the polynomial x2−3 and verify the relationship between the zeroes and the coefficients.
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Step-by-Step Solution
Step 1: Find the zeroes of the polynomial
To find the zeroes of the polynomial, we set the polynomial equal to zero. This allows us to solve for the values of x that make the polynomial zero.
Step 2: Factor the polynomial using the identity a2−b2=(a−b)(a+b)
We recognize the polynomial x2−3 as a difference of squares. We can rewrite 3 as (3)2, so the expression becomes x2−(3)2. Applying the difference of squares identity, a2−b2=(a−b)(a+b), we get (x−3)(x+3). Setting this factored form to zero allows us to find the roots.
Step 3: Solve for x
For the product of two factors to be zero, at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for x. This gives us the two zeroes of the polynomial.
Step 4: Identify coefficients and zeroes
The given polynomial is x2−3, which can be written as ax2+bx+c. By comparing, we find the coefficients a=1, b=0, and c=−3. The zeroes we found are α=3 and β=−3.
Step 5: Verify the sum of zeroes
According to the relationship between zeroes and coefficients, the sum of the zeroes (α+β) should be equal to -b/a. We calculate both values and confirm they are equal, verifying the relationship for the sum of zeroes.
Step 6: Verify the product of zeroes
Similarly, the product of the zeroes (αβ) should be equal to c/a. We calculate both values and confirm they are equal, verifying the relationship for the product of zeroes.