Constructing a Quadratic Polynomial with Specific Zero Properties

Step 1: Understanding Vieta’s Formulas
For a quadratic polynomial ax² + bx + c with zeros p and q:
– Sum of zeros: p + q = -b/a
– Product of zeros: p * q = c/a

Given Conditions:
– Sum of zeros = -4
– Product of zeros = 3

Step 2: Analyzing the Coefficients
Let’s consider a standard quadratic form: x² + 4x + c

Checking Sum of Zeros:
– p + q = -4
– This means the coefficient of x must be -4

Checking Product of Zeros:
– p * q = 3
– This means the constant term must be 3

Step 3: Verification
The polynomial becomes: x² – 4x + 3

Mathematical Verification:
Let’s find the zeros using the quadratic formula:
x = [4 ± √(16 – 4(1)(3))] / 2(1)
= [4 ± √(16 – 12)] / 2
= [4 ± √4] / 2
= [4 ± 2] / 2

Zeros are:
– p = (4 + 2)/2 = 3
– q = (4 – 2)/2 = 1

Checking Conditions:
– Sum of zeros: 3 + 1 = 4 ✓
– Product of zeros: 3 * 1 = 3 ✓

Key Insights:
– Vieta’s formulas provide a powerful way to relate
zeros to polynomial coefficients
– We can construct polynomials by understanding
the relationships between zeros and coefficients

Conclusion:
The polynomial that satisfies the given conditions is x² – 4x + 3.

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