Dada una función real f(x), se define una raíz de f como el (los) valor (es) de x tal(es) que cumplen f(x) = 01?

<img src="https://tex.z-dn.net/?f=f%28x%29%3Dx%5E3-2ax%5E2-37x%2B30a" />

Como x = 3a es una raíz, f(3a) = 0 :

<img src="https://tex.z-dn.net/?f=f%283a%29%3D%283a%29%5E3-2a%283a%29%5E2-37%283a%29%2B30a%5C%5C%5C%5C%0A0%3D27a%5E3-2a%5Ccdot9a%5E2-111a%2B30a%5C%5C%5C%5C%0A27a%5E3-18a%5E3-81a%3D0%5C%5C%5C%5C%0A9a%5E3-81a%3D0%5C%5C%5C%5C%0A9a%28a%5E2-9%29%3D0%5C%5C%5C%5C%0A9a%28a%2B3%29%28a-3%29%3D0" />

Entonces :

<img src="https://tex.z-dn.net/?f=9a%3D0~~o~~a%2B3%3D0~~o~~a-3%3D0%5C%5C%5C%5C%0Aa%3D0~~~o~~a%3D-3~~~~~o~~a%3D3" />

Según el enunciado, <img src="https://tex.z-dn.net/?f=3a%3E0%5Ciff%20a%3E0" />0 \ iff a>0" alt = "3a>0 \ iff a>0" align = "absmiddle" class = "latex - formula">.

Por lo tanto, <img src="https://tex.z-dn.net/?f=%5Cboxed%7Ba%3D3%7D" />

Tenemos que <img src="https://tex.z-dn.net/?f=3a%3D3%5Ccdot3%3D9" /> es una raíz.

Por consiguiente, (x - 9) es un factor de f(x) :

<img src="https://tex.z-dn.net/?f=f%28x%29%3Dx%5E3-6x%5E2-37x%2B90%5C%5C%5C%5C%0A%28x-9%29%28x%5E2%2Bc_1x%2Bc_0%29%3Dx%5E3-6x%5E2-37x%2B90%5C%5C%5C%5C%0Ax%5E3%2B%5Cunderbrace%7B%28c_1-9%29%7D_%7B%28iii%29%7Dx%5E2%2B%5Cunderbrace%7B%28c_0-9c_1%29%7D_%7B%28ii%29%7Dx%5Cunderbrace%7B-9c_0%7D_%7B%28i%29%7D%3Dx%5E3%5Cunderbrace%7B-6%7D_%7B%28iii%29%7Dx%5E2%5Cunderbrace%7B-37%7D_%7B%28ii%29%7Dx%2B%5Cunderbrace%7B90%7D_%7B%28i%29%7D%5C%5C%5C%5C%5C%5C%0A%28i%29%3A~-9c_0%3D90%5Ciff%20c_0%3D-10%5C%5C%5C%5C%0A%28iii%29%3A~c_1-9%3D-6%5Ciff%20c_1%3D3%5C%5C%5C%5C%0A%28ii%29%3A~c_0-9c_1%3D-37%5CLongrightarrow%20-10-9%5Ccdot3%3D-37%5Cto%20Ok%21%5C%5C%5C%5C%5C%5C%0Af%28x%29%3D%28x-9%29%28x%5E2%2B3x-10%29" />

Si f(x) = 0 y x≠9 :

<img src="https://tex.z-dn.net/?f=x%5E2%2B3x-10%3D0%5C%5C%5C%5C%5C%5C%0A%5CDelta%3Db%5E2-4ac%5C%5C%5C%5C%0A%5CDelta%3D3%5E2-4%5Ccdot1%5Ccdot%28-10%29%5C%5C%5C%5C%0A%5CDelta%3D9%2B40%5C%5C%5C%5C%0A%5CDelta%3D49%3D7%5E2%5C%5C%5C%5C%5C%5C%0Ax%3D%5Cdfrac%7B-b%5Cpm%5Csqrt%7B%5CDelta%7D%7D%7B2a%7D%3D%5Cdfrac%7B-3%5Cpm%5Csqrt%7B7%5E2%7D%7D%7B2%5Ccdot1%7D%3D%5Cdfrac%7B-3%5Cpm7%7D%7B2%7D%5C%5C%5C%5C%5C%5C%0Ax_1%3D%5Cdfrac%7B-3%2B7%7D%7B2%7D%5Ciff%20x_1%3D%5Cdfrac%7B4%7D%7B2%7D%5Ciff%20%5Cboxed%7Bx_1%3D2%7D%5C%5C%5C%5C%0Ax_2%3D%5Cdfrac%7B-3-7%7D%7B2%7D%5Ciff%20x_2%3D%5Cdfrac%7B-10%7D%7B2%7D%5Ciff%20%5Cboxed%7Bx_2%3D-5%7D" />

Las otras raíces son x = 2 y x = - 5.