1.Rozłóż na czynniki:
a) 4x³-8x=
b)2x do 4-5x³-16x+40=
2.Liczba -3 jest pierwiastkiem wielomianu W(x)=x³+mx²-25x-75.Znajdź wartość m i pozostałe pierwiastki tego wielomianu.
Proszę o dokładne rozwiązanie i rozpisanie.
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1. a)![4x^3-8x=4x(x^2-2)=4x(x-\sqrt2)(x+\sqrt2) 4x^3-8x=4x(x^2-2)=4x(x-\sqrt2)(x+\sqrt2)](https://tex.z-dn.net/?f=4x%5E3-8x%3D4x%28x%5E2-2%29%3D4x%28x-%5Csqrt2%29%28x%2B%5Csqrt2%29)
b)![2x^4-5x^3-16x+40=x^3(2x-5)-8(2x-5)=\\=(2x-5)(x^3-8)=(2x-5)(x-2)(x^2+2x+4) 2x^4-5x^3-16x+40=x^3(2x-5)-8(2x-5)=\\=(2x-5)(x^3-8)=(2x-5)(x-2)(x^2+2x+4)](https://tex.z-dn.net/?f=2x%5E4-5x%5E3-16x%2B40%3Dx%5E3%282x-5%29-8%282x-5%29%3D%5C%5C%3D%282x-5%29%28x%5E3-8%29%3D%282x-5%29%28x-2%29%28x%5E2%2B2x%2B4%29)
2.![W(-3)=(-3)^3+m(-3)^2-25\cdot(-3)-75=\\=-27+9m+75-75=0\\9m=27\\m=3\\W(x)=x^3+3x^2-25x-75=(x+3)(x^2-25)=\\=(x+3)(x-5)(x+5) W(-3)=(-3)^3+m(-3)^2-25\cdot(-3)-75=\\=-27+9m+75-75=0\\9m=27\\m=3\\W(x)=x^3+3x^2-25x-75=(x+3)(x^2-25)=\\=(x+3)(x-5)(x+5)](https://tex.z-dn.net/?f=W%28-3%29%3D%28-3%29%5E3%2Bm%28-3%29%5E2-25%5Ccdot%28-3%29-75%3D%5C%5C%3D-27%2B9m%2B75-75%3D0%5C%5C9m%3D27%5C%5Cm%3D3%5C%5CW%28x%29%3Dx%5E3%2B3x%5E2-25x-75%3D%28x%2B3%29%28x%5E2-25%29%3D%5C%5C%3D%28x%2B3%29%28x-5%29%28x%2B5%29)
Zatem pierwiastki wielomianu to -5, -3 i 5