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Find a polynomial function of degree 4 with -1 as a zero of multiplicity 3 and 0 as a zero of multiplicity 1.

Find a polynomial function of degree 4 with -1 as a zero of multiplicity 3 and 0 as-example-1
User Ethry
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Explanation:

a polynomial of degree 4 must have 4 zeroes.

when we know these zeroes, we can define the polynomial by the factors defining these zeroes.

after all, a multiplication of several factors can only be zero, if at least one of the factors is zero.

we have 3 zeroes at x = -1. and 1 zero at x = 0.

so,

f(x) = (x + 1)(x + 1)(x + 1)x

you see ? this will only be zero, if x = -1 or x = 0. and -1 is built in as zero 3 times.

let's do the multiplications :

(x + 1)(x + 1) = x² + 2x + 1

(x + 1)x = x² + x

(x² + 2x + 1)(x² + x) = x⁴ + x³ + 2x³ + 2x² + x² + x =

= x⁴ + 3x³ + 3x² + x

f(x) = 1x⁴ + 3x³ + 3x² + 1x

User Gecco
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