F(x) = 4x^3+...+ 9
list all possible rational zeroes of the function

The Rational Root Theorem says any rational zeros of f must be a fraction with a factor of the constant (9 here) in the numerator and a factor of the leading coefficient (4 here) in the denominator.   We have to include negative factors.

Factors of 9:  -9, -3, -1, 1, 3, 9

Factors of 4: 1, 2, 4

We only have to consider negative numbers for the numerator to get all the possible values.

We just make the list, going through all the numerators one denominator at a time.

Possible rational zeros, i.e. possible rational values for x such that f(x)=0 are

-9, -3, -1, 1, 3, 9,   -9/2, -3/2, -1/2, 1/2, 3/2, 9/2,   -9/4, -3/4, -1/4, 1/4, 3/4, 9/4

step-by-step explanation:

this is not possible to solve with the info given.   you need at least an one angle somewhere.

m∠cdf = 43° , m∠gdf = 114°, m∠gde = 23°, m∠edh = 23° , m∠cdh = 157°

step-by-step explanation:

since, it is given that de bisects the angle gdh ⇒ de divides the angle gdh into two equal angles.

⇒ m∠gde = m∠edh

⇒ 8·x   - 1 = 6·x + 5

⇒ x = 3

so, m∠gde = 8·x - 1 = 23°

and m∠edh = 23°

now, it is given ∠cde is an straight angle so, m∠cde = 180°

⇒ m∠cdf + m∠fdg + m∠gde = 180°

⇒ 43° + m∠fdg + 23° = 180°

⇒ m∠fdg = 114°

again, m∠cde = m∠cdh + m∠edh = 180°

m∠cdh + 23° = 180°

⇒ m∠cdh = 157°