Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

According to Descartes' Rule of Signs, can the polynomial function have exactly $5$ positive real zeros, including any repeated zeros? Choose your answer based on the rule only. $\newline$ $g(x) = x^5 - 9x^4 - 4x^2 + 2x + 8$ $\newline$ Choices: $\newline$ (A)yes $\newline$ (B)no

View step-by-step help

View step-by-step help

Descartes' Rule of Signs

Full solution

Q. According to Descartes' Rule of Signs, can the polynomial function have exactly

5

positive real zeros, including any repeated zeros? Choose your answer based on the rule only.

\newline

g(x) = x^5 - 9x^4 - 4x^2 + 2x + 8

\newline

Choices:

\newline

(A)yes

\newline

(B)no

Count Sign Changes: Count the number of sign changes in the coefficients of $g(x) = x^5 - 9x^4 - 4x^2 + 2x + 8$ . Coefficients: $1, -9, 0, -4, 2, 8$ . Sign changes: $1$ to $-9$ (one change), $-4$ to $2$ (second change), $2$ to $8$ (no change). Total sign changes: $2$ .
Descartes' Rule of Signs: According to Descartes' Rule of Signs, the number of positive real zeros is equal to the number of sign changes or less than that by an even number. $\newline$ So, $g(x)$ can have $2$ or $0$ positive real zeros.
Check Positive Zeros: Check if $g(x)$ can have exactly $5$ positive real zeros. $\newline$ Since $g(x)$ can have at most $2$ positive real zeros, it cannot have exactly $5$ positive real zeros.

More problems from Descartes' Rule of Signs

Question

Find the degree of this polynomial.

\newline

`5b^{10} + 3b^3`

\newline

Posted 2 months ago

Question

\newline

(9a + 5) - (2a + 4)

\newline

Posted 2 months ago

Question

Find the product. Simplify your answer.

\newline

-3v^2(v^2 - 9)

\newline

Posted 1 month ago

Question

Find the product. Simplify your answer.

\newline

(v - 3)(4v + 1)

\newline

Posted 2 months ago

Question

Find the square. Simplify your answer.

\newline

(3y + 2)^2

\newline

Posted 2 months ago

Question

Divide. If there is a remainder, include it as a simplified fraction.

\newline

(24t^2 + 36t) \div 6t

\newline

Posted 2 months ago

Question

Is the function

q(x) = x^6 - 9

even, odd, or neither?

\newline

\newline

\text{[[even][odd][neither]]}

Posted 2 months ago

Question

Find the product. Simplify your answer.

\newline

-3q^2(-3q^2 + q)

\newline

Posted 2 months ago

Question

Find the product. Simplify your answer.

\newline

(r + 3)(4r + 2)

\newline

Posted 2 months ago

Question

Find the roots of the factored polynomial.

\newline

(x + 7)(x + 4)

\newline

Write your answer as a list of values separated by commas.

\newline

x =

Posted 2 months ago