Simplify the expression by using a double-angle formula. 2 cos^2 4 theta -1

1 answer:

3 0

You are given 2cos² (4θ - 1). You are asked to s<span>implify the expression by using a double-angle formula.

Apply the double angle of the cosine function
</span>cos2θ = 2cos² θ - 1

Rearranging the expression and then substituting them
2cos² θ = cos2θ + 1
2cos² (4θ - 1) = cos 2(4θ - 1) + 1
cos2(4θ - 1) + 1

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According to Descartes's rule of sign, how many possible positive and negative roots are there for the equation

damaskus [11]

<h2>Descartes Rule of Sign</h2><h3>Answer:</h3>

There are $3$ Real Roots for the Equation

<h3>Step-by-step explanation:</h3>

According to Descartes Rule of Sign, The maximum number of real roots a polynomial equation has is the number of times the sign of term changes from a higher power term to a lower power term.

In the given equation, $0=-4x^6-3x^5+2x^2-4x+1$ , we can already see that from $\red -4x^6$ to $\red -3x^5$ we can see that the sign didn't change so there might be no real roots. From $\red -3x^5$ to $\blue +2x^2$ , we can see that the signs change so we now know that there might be $1$ real root for equation. From $\blue +2x^2$ to $\red -4x$ , the signs changed so there might be $2$ real roots. From $\red -4x$ to $\blue +1$ , it changed again so there might be $3$ real roots. There are no other term with lesser power to compare for $\blue +1$ so we are left with $3$ real roots.