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.

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.

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.

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.

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.

Find the zeroes of the polynomial $x^2 - 3$ and verify the relationship between the zeroes and the coefficients.

Answer: The zeroes of the polynomial $x^2 - 3$ are $\sqrt{3}$ and $-\sqrt{3}$ . Verification: Sum of zeroes: $\sqrt{3} + (-\sqrt{3}) = 0$ . $-b/a = -0/1 = 0$ . (Verified) Product of zeroes: $(\sqrt{3})(-\sqrt{3}) = -3$ . $c/a = -3/1 = -3$ . (Verified)

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 $a^2 - b^2 = (a - b)(a + b)$

We recognize the polynomial $x^2 - 3$ as a difference of squares. We can rewrite $3$ as $(\sqrt{3})^2$ , so the expression becomes $x^2 - (\sqrt{3})^2$ . Applying the difference of squares identity, $a^2 - b^2 = (a - b)(a + b)$ , we get $(x - \sqrt{3})(x + \sqrt{3})$ . 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 $x^2 - 3$ , which can be written as $ax^2 + bx + c$ . By comparing, we find the coefficients $a=1$ , $b=0$ , and $c=-3$ . The zeroes we found are $\alpha = \sqrt{3}$ and $\beta = -\sqrt{3}$ .

Step 5: Verify the sum of zeroes

According to the relationship between zeroes and coefficients, the sum of the zeroes ( $\alpha + \beta$ ) 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 ( $\alpha \beta$ ) should be equal to $c/a$ . We calculate both values and confirm they are equal, verifying the relationship for the product of zeroes.

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