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Find the equation of a polynomial with given zeros: −1 multiplicity 2, −3,−5

User Syncere
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Explanation:

a "zero" is a value of x that creates 0 as functional value of the polynomial.

for this we can use the fact that the product

0 × something = 0

for whatever that "something" is.

so, we write the polynomial expression as a product of factors. each factor is a small term that turns to 0 for a given value of x :

x + 1 = 0 for x = -1

(x + 1)² = 0 for x = -1 and multiplicity 2 (this zero counts as 2 zeroes)

x + 3 = 0 for x = -3

x + 5 = 0 for x = -5

and our polynomial is

f(x) = (x + 1)²(x + 3)(x + 5) =

= (x² + 2x + 1)(x + 3)(x + 5) =

= (x² + 2x + 1)(x² + 5x + 3x + 15) =

= (x² + 2x + 1)(x² + 8x + 15) =

= x⁴ + 8x³ + 15x² + 2x³ + 16x² + 30x + x² + 8x + 15 =

= x⁴ + 10x³ + 32x² + 38x + 15

User Chrmue
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