How do you write a polynomial function in standard form with Zero: 5, multiplicity 3?

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How do you write a polynomial function in standard form with Zero: 5, multiplicity 3?

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Answer 1 · 2016-07-15T17:43:46

$x^{3} - 15 x^{2} + 75 x - 125$ $x^3-15x^2+75x-125$

Explanation:

A zero, say $a$ $a$ of a polynomial $f (x)$ $f(x)$ is one for which $f (a) = 0$ $f(a)=0$ .

Let us assume that zeros are ${a, b, c, d}$ ${a,b,c,d}$ , then it is apparent that such a polynomial could be $(x - a) (x - b) (x - c) (x - d)$ $(x-a)(x-b)(x-c)(x-d)$ .

As we need to write a polynomial with zero $5$ $5$ with multiplicity $3$ $3$ , the polynomial is

${(x - 5)}^{3}$ $(x-5)^3$ and using identity ${(x - a)}^{3} = x^{3} - 3 a x^{2} + 3 a^{2} x - a^{3}$ $(x-a)^3=x^3-3ax^2+3a^2x-a^3$ , this can be expanded
as

$x^{3} - 3 \times x^{2} \times 5 + 3 \times x \times 5^{2} - 5^{3}$ $x^3-3×x^2×5+3×x×5^2-5^3$ or

$x^{3} - 15 x^{2} + 75 x - 125$ $x^3-15x^2+75x-125$ .

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How do you write a polynomial function in standard form with Zero: 5, multiplicity 3?

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