Three roots of a fifth degree polynomial function f(x) are –2, 2, and 4 + i. which statement describes the number and nature of all roots for this function? f(x) has two real roots and one imaginary root. f(x) has three real roots. f(x) has five real roots. f(x) has three real roots and two imaginary roots.

Because the polynomial is of the fifth degree, we expect five roots.  If 4 + i is one of those roots, there must be a corresponding 4 - i.  We can conclude that there are two complex roots and three real roots.   The answer choice that involves the word "imaginary" cannot be correct; rather, we must use the word "complex" root to describe roots that have both real and imaginary parts.  "f(x) has three real roots" is the correct answer here.

d

step-by-step explanation:

hello and welcome!

so, when we see the words "proportional" along with a table, we need to know that we're looking numbers that divide by the same thing.

looking at this table, we simply have to do a trial and error, and divide to see which numbers have the same quotient. if you look at choice a & c, they both have equal quotients for three of the numbers, with one exception, so those can't be the answer. looking at b, there is no similarity, so that can't be it. then, we see d; if you divide all the numbers in the table, you'll see that they all divide by two, which is a proportional relationship, and our answer!

also, always remember that the x will always be the number that goes into y, and somewhere behind this table, is a formula. for this specific table, the formula would be:

since our x-value is being divided by two to get the y-value. and if they were to ask for more numbers in the series, this is the formula you would use to find them.

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