Consider p(x)=3x^3+ax+5a, a is an element of the integers. The polynomial p(x) leaves a remainder of -7 when divided by (x-a). Show that only one value of a satisfies the above condition and state it’s value.

By the remainder theorem,

p(a) = 3a³ + a² + 5a = -7

so that

3a³ + a² + 5a + 7 = 0

Since you know a is an integer, you can try to factorize this, or use the rational root theorem to generate a list of potential rational roots, and from there you'd select the integer roots. The rational ones are of the form m/n, where m divides the constant term 7 and n divides the leading coefficient 3. They are

a = ± 1 or ± 1/3 or ± 7 or ±7/3

so there are 4 integer candidates,

a = -1 or 1 or -7 or 7

Check the value of f(a) = 3a³ + a² + 5a + 7 at each of these:

a = -1   →   f(a) = 0

a = 1   →   f(a) = 16

a = -7   →   f(a) = -1008

a = 7   →   f(a) = 1120

and only a = -1 is a root of 3a³ + a² + 5a + 7, so that's your answer. In fact,

3a³ + a² + 5a + 7 = (a + 1) (3a² - 2a + 7)

c. 27 units.

step-by-step explanation:

we are told that in circle a, ∠bae ≅ ∠dae.

we can see from our given diagram that in and ;

, as these are radii of our given circle.

therefore, by sas congruence postulate.

hence, side be will be equal to side de as corresponding parts of congruent triangles are congruent. so we can set an equation to find the value of x as:

now let us substitute x=17 in the expression for the length of be.

therefore, length of be will be 27 units and option c is the correct choice.

the 8 in 8,420 is in the thousands place so it's value is 8 thousand.

the 8 in 6,870 is in the hundreds place so it's value is 8 hundred.

divide 8000 by 800 to find the answer:

8000 / 800 = 10

the 8 in 8,420 is 10 times greater than the 8 in 6,870