Question

according to descartes rule of signs the polynomial

( p ( x ) = x ^ { 4 } - 6 x ^ { 3 } + 4 x ^ { 2 } + x - 5 )

has changes in sign, which means that the polynomial has

4 2 1 3

three positive real roots or two positive real roots two positive real roots or one positive real root three positive real roots or one positive real root

two positive real roots or zero positive real roots

write the simplest polynomial function with the given roots:

( { - 3,2,4 } )

( p ( x ) = x ^ { 3 } + x ^ { 2 } + x + )

-1 2 -3

-10 -6 -4

24 -24 12

Explanation:

First Problem (Descartes' Rule of Signs)

Step1: List coefficients of \(P(x)\)

Coefficients: \(1, -6, 4, 1, -5\)

Step2: Count sign changes

Changes: \(1\to-6\) (1), \(-6\to4\) (2), \(1\to-5\) (3) → Total 3

Step3: Apply Descartes' Rule

Positive roots: 3 or \(3-2=1\)

Step1: Write factored form

Roots \(-3,2,4\): \(P(x)=(x+3)(x-2)(x-4)\)

Step2: Expand \((x+3)(x-2)\)

\(x^2 + 3x -2x -6 = x^2 + x -6\)

Step3: Multiply by \((x-4)\)

\((x^2 + x -6)(x-4) = x^3 -4x^2 +x^2 -4x -6x +24 = x^3 -3x^2 -10x +24\)

Answer:

3; Three positive real roots or one positive real root

Second Problem (Polynomial with Given Roots)