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F(x)=x^3-7x^2+25x-175 and f(7)=0

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4 votes

Si nous savons que f(7) = 0 alors,

x - 7 est un facteur de la fonction f(x).

on substitue x = 7 dans la fonction

f(7) = 7^3 - 7(7)^2 + 25(7) - 175 = 0

donc (x - 7) est l’un des facteurs de la fonction.

Maintenant, on utilise division polynomiale ou la division synthétique pour avoir les autres facteurs. Si on divise la fonction f(x) par (x - 7), on obtient:

x^2 - 4x + 25

Du coup peut dire que :

f(x) = (x - 7)(x^2 - 4x + 25)

Le terme quadratique, x^2 - 4x + 25, n’a pas de racines réelles car le discriminant est négatif. Par conséquent, nous pouvons écrire la factorisation de f(x) en termes de nombres complexes comme:

f(x) = (x - 7)(x - 2 + 5i)(x - 2 - 5i)

la factorisation complète de f(x) est:

f(x) = (x - 7)(x - 2 + 5i)(x - 2 - 5i)

User Marcosc
by
7.8k points
6 votes

If we know that f(7) = 0, we can say that:

x - 7 is a factor of the function f(x).

This is because when we substitute x = 7 into the function, we get:

f(7) = 7^3 - 7(7)^2 + 25(7) - 175 = 0

This means that (x - 7) is one of the factors of the function.

Now we can use polynomial division or synthetic division to find the other factors. If we divide the function f(x) by (x - 7), we obtain:

x^2 - 4x + 25

Therefore, we can say that:

f(x) = (x - 7)(x^2 - 4x + 25)

The quadratic term, x^2 - 4x + 25, has no real roots because the discriminant is negative. Therefore, we can write the factorization of f(x) in terms of complex numbers as:

f(x) = (x - 7)(x - 2 + 5i)(x - 2 - 5i)

Thus, the complete factorization of f(x) is:

f(x) = (x - 7)(x - 2 + 5i)(x - 2 - 5i)

User Olivier Gourment
by
7.9k points

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