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gurjot kaur gurjot kaur
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3 years ago
 Prove that x4 + 2x3 −x = 1 has at least two real roots.
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Educator
3 years ago
Prove that x4 + 2x3 −x = 1 has at least two real roots.

Have you tried using Descartes’ Rule of Signs?

x4 + 2x3 −x = 1

x^4 + 2x^3 + 0x^2 - x - 1 = 0

The number of positive real zeros of f is equal to the number of variations in sign of f(x) or is less than the number of variations in sign of f(x) by a positive even integer. If f(x) has one variation in sign, then f has exactly one positive real zero.

The number of negative real zeros of f is equal to the number of variations in sign of f(-x) or is less than the number of variations in sign of f(-x) by a positive even integer. If f(-x) has one variation in sign, then f has exactly one negative real zero



You may also need to apply:

Fundamental Theorem of Algebra: Every polynomial function of degree n>=1 has at least one complex zero.
Number of Zeros Theorem: Every polynomial of degree n has n complex zeros provided each zero of multiplicity greater than 1 is counted accordingly.

PS: I'll be answering the rest of your questions tomorrow... My shift if done for the day Wink Face
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