Find the zeroes of the quadratic polynomial x2+7x+10, and verify the relationship between the zeroes and the coefficients.
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Step-by-Step Solution
Step 1: Factor the quadratic polynomial
To find the zeroes of the quadratic polynomial, we need to set the polynomial equal to zero and solve for x. We can factor the quadratic expression by finding two numbers that multiply to 10 and add up to 7. These numbers are 2 and 5.
Step 2: Factor the expression
We rewrite the middle term 7x as 2x + 5x. Then, we factor by grouping. We take out the common factor x from the first two terms and 5 from the last two terms. This results in the factored form (x+2)(x+5)=0.
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. The zeroes are x=−2 and x=−5.
Step 4: Identify coefficients
For a general quadratic polynomial ax2+bx+c, we identify the coefficients. In our case, for x2+7x+10, we have a=1, b=7, and c=10.
Step 5: Verify sum of zeroes
The sum of the zeroes of a quadratic polynomial ax2+bx+c is given by the formula −ab. We calculate the sum of the zeroes we found, −2+(−5)=−7, and compare it with −ab. Both values are −7, so the relationship is verified.
Step 6: Verify product of zeroes
The product of the zeroes of a quadratic polynomial ax2+bx+c is given by the formula ac. We calculate the product of the zeroes we found, (−2)×(−5)=10, and compare it with ac. Both values are 10, so the relationship is verified.