Final answer:
To write a polynomial with given zeros of -2/5, 5/2, and 3, you turn each zero into a factor, multiply these factors together, and expand. This will result in a polynomial equation with integer coefficients that meets the given conditions.
Step-by-step explanation:
The student is looking to write a polynomial with integer coefficients that has zeros at -2/5, 5/2, and 3. Polynomials are algebraic expressions composed of variables and coefficients. To find such a polynomial, each zero corresponds to a factor of the polynomial. For irrational or fractional roots, the factors are typically written in a way that will ultimately lead to integer coefficients when multiplied out.
Given the zeros of -2/5, 5/2, and 3, the corresponding factors of the polynomial are (5x + 2), (2x - 5), and (x - 3). Multiplying the factors together, we get:
Step 1: (5x + 2)(2x - 5)(x - 3)
Step 2: First, multiply (5x + 2)(2x - 5) to get 10x2 - 25x + 4x - 10.
Step 3: Combine like terms to get 10x2 - 21x - 10.
Step 4: Now, multiply (10x2 - 21x - 10)(x - 3).
Step 5: Distribute each term in the first polynomial across the second polynomial.
Final equation: After expanding and combining like terms, you will end up with a cubic polynomial with integer coefficients that satisfies the given conditions, such as: 10x3 - 51x2 + 63x - 30.
Remember, the exact coefficients will depend on how the factors are multiplied and combined, and there may be multiple correct answers as long as they satisfy the specified zeros and have integer coefficients.