Find the zeroes of the quadratic polynomial x2 - 2x - 8 and verify the relationship between the zeroes and the coefficients.
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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×(−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 ax2+bx+c, we identify the coefficients a, b, and c from our given polynomial x2−2x−8. Here, a=1, b=−2, and c=−8.
Step 5: Verify sum of zeroes
Let the zeroes be α=4 and β=−2. The sum of the zeroes is α+β. 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 αβ. 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.