Verify that 3,−1,−31 are the zeroes of the cubic polynomial p(x)=3x3−5x2−11x−3, and then verify the relationship between the zeroes and the coefficients.
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Step-by-Step Solution
Step 1: Evaluate p(x) at x=3
To verify that a number is a zero of a polynomial, we substitute that number into the polynomial and check if the result is zero. We will start by substituting x=3 into the polynomial p(x)=3x3−5x2−11x−3.
Step 2: Calculate p(3)
After substituting x=3, we perform the arithmetic operations: 33=27, 32=9. Then, we calculate the products and sums to find the value of p(3). Since p(3)=0, x=3 is indeed a zero of the polynomial.
Step 3: Evaluate p(x) at x=−1
Next, we substitute x=−1 into the polynomial p(x) to check if it is also a zero. We replace every x in the expression with −1.
Step 4: Calculate p(−1)
We calculate (−1)3=−1 and (−1)2=1. Then we perform the multiplications and additions. Since p(−1)=0, x=−1 is also a zero of the polynomial.
Step 5: Evaluate p(x) at x=−31
Finally, we substitute x=−31 into the polynomial p(x) to verify the last given zero. We carefully substitute the fractional value into the expression.
Step 6: Calculate p(−31)
We calculate (−31)3=−271 and (−31)2=91. Then we simplify the expression by finding a common denominator for the fractions. Since p(−31)=0, x=−31 is also a zero of the polynomial. Thus, all three given values are verified zeros.
Step 7: Identify coefficients of p(x)
Now we will verify the relationship between the zeroes and the coefficients. A general cubic polynomial is given by ax3+bx2+cx+d. By comparing this with p(x)=3x3−5x2−11x−3, we can identify the coefficients a, b, c, d.
Step 8: Assign values to zeroes
Let the zeroes be α, β, and γ. We assign the given verified zeroes to these variables for convenience in applying the relationships.
Step 9: Verify sum of zeroes
The relationship for the sum of the zeroes of a cubic polynomial is α+β+γ=−ab. We calculate the sum of our zeroes and compare it with -b/a using the coefficients identified earlier. Both values match, verifying this relationship.
Step 10: Verify sum of products of zeroes taken two at a time
The relationship for the sum of the products of the zeroes taken two at a time is αβ+βγ+γα=ac. We compute this sum using our zeroes and compare it with c/a. Both results are equal, confirming this relationship.
Step 11: Verify product of zeroes
The relationship for the product of the zeroes of a cubic polynomial is αβγ=−ad. We calculate the product of our zeroes and compare it with -d/a. The values match, which verifies the final relationship.