We can factor the quadratic equation by splitting the middle term. We look for two numbers that multiply to AC = 1 × (-8) = -8 and add up to B = -2. These numbers are -4 and 2. Then, we group the terms and factor out common factors.

We can factor the quadratic equation by splitting the middle term. We look for two numbers that multiply to AC = 1 × (-8) = -8 and add up to B = -2. These numbers are -4 and 2. Then, we group the terms and factor out common factors.

Find the zeroes of the quadratic polynomial $x^2 - 2x - 8$ and verify the relationship between the zeroes and the coefficients.

Answer: The zeroes of the quadratic polynomial $x^2 - 2x - 8$ are $4$ and $-2$ . The relationships between the zeroes and coefficients are verified as: Sum of zeroes ( $4 + (-2) = 2$ ) equals $-b/a$ ( $-(-2)/1 = 2$ ), and Product of zeroes ( $4 \times (-2) = -8$ ) equals $c/a$ ( $-8/1 = -8$ ).

Step-by-step solution

Step 1: Set the polynomial to zero

To find the zeroes of a quadratic polynomial, we set the polynomial equal to zero. This converts the polynomial into a quadratic equation, whose solutions are the zeroes of the polynomial.

Step 2: Factor the quadratic equation

We can factor the quadratic equation by splitting the middle term. We look for two numbers that multiply to $AC = 1 \times (-8) = -8$ and add up to $B = -2$ . These numbers are $-4$ and $2$ . Then, we group the terms and factor out common factors.

Step 3: Find the zeroes

For the product of two factors to be zero, at least one of the factors must be zero. So, we set each factor equal to zero and solve for $x$ to find the zeroes of the polynomial.

Step 4: Identify coefficients

For a general quadratic polynomial $ax^2 + bx + c$ , we identify the coefficients $a$ , $b$ , and $c$ from our given polynomial $x^2 - 2x - 8$ . Here, $a=1$ , $b=-2$ , and $c=-8$ .

Step 5: Verify sum of zeroes

Let the zeroes be $\alpha = 4$ and $\beta = -2$ . The sum of the zeroes is $\alpha + \beta$ . We compare this to the formula for the sum of zeroes, which is $-b/a$ . Since both values are equal to $2$ , the relationship is verified.

Step 6: Verify product of zeroes

The product of the zeroes is $\alpha \beta$ . We compare this to the formula for the product of zeroes, which is $c/a$ . Since both values are equal to $-8$ , the relationship is verified.

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Find the zeroes of the quadratic polynomial x2−2x−8x^2 - 2x - 8x2−2x−8 and verify the relationship between the zeroes and the coefficients.