So, the polynomial with the require properties is x⁵ + x⁴ - 3x³ - 5x² - 2x
<h3>What is a polynomial?</h3>
A polynomial is an expression of the form aₙxⁿ + aₙ₋₁xⁿ⁻¹ + aₙ₋₂xⁿ⁻² + ... + a₂x² + a₁x + a₀ where aₙ are coefficients and n ≥ 2.
<h3>How to construct the given polynomail?</h3>
Since we are given a polynomial with the following properties: fifth degree, −1 is a zero of multiplicity 3, 2 is the only other zero.
We have that it has zero at
⇒ x + 1 = 0 and x - 2 = 0
⇒ (x + 1) and (x - 2) are factors of the polynomial
We are told that the polynomial has a zero of -1 with multiplicity of 3. This implies that the factor (x + 1) is repeated 3 times. That is (x + 1)(x + 1)(x + 1)
So, multiplying all four factors we have a polynomial of fourth degree as p(x) = (x + 1)(x + 1)(x + 1)(x - 2)
= (x² + 2x + 1)(x + 1)(x - 2)
= (x³ + 3x² + 3x + 1)(x - 2)
= x⁴ + x³ - 3x² - 5x - 2
To obtain a polynomial of degree 5, we multiply through by x, so we have
g(x) =xp(x)
= x(x⁴ + x³ - 3x² - 5x - 2)
= x⁵ + x⁴ - 3x³ - 5x² - 2x
So, the required polynomial is x⁵ + x⁴ - 3x³ - 5x² - 2x
Learn more about polynomials here:
brainly.com/question/2833285
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