Find the value of for which the polynomial(5 −^2^3 +2+−3)isamultipleof(−).
Answers
Answer:
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Answer:
Step-by-step explanation:
Find all the zeroes of: y = 2x5 + 3x4 – 30x3 – 57x2 – 2x + 24
First, I'll apply the Rational Roots Test—
Wait. Actually, the first thing I'll do is apply a trick I've learned. First, I'll check to see if either x = 1 or x = –1 is a root.
(These are the simplest roots to test for. This isn't an "official" first step, but it can often be a timesaver, because (a) it's amazing how often one of these is a zero, and (b) you can just look at the powers and the numbers to figure out if either works, because of how 1 and –1 simplify.)
When x = 1, the polynomial evaluates as:
2 + 3 – 30 – 57 – 2 + 24 = –60
This isn't equal to zero, so x = 1 isn't a root. But when x = –1, I get:
–2 + 3 + 30 – 57 + 2 + 24 = 0
This time, it did equal zero, so now I know that x = –1 is a root, and I can take "prove" this (in my hand-in work) by using synthetic division:
synthetic division by x = –1; –1 is to the left; the top line inside the division is 2 3 –30 –57 –2 24; the second line inside the division, after the initial empty slot, is –2 –1 31 26 –24; below the second line is a horizontal "equals" bar; the last line is 2 1 –31 –26 24 0
The last line of this division shows me with the new, smaller polynomial equation I'm working with now:
2x4 + x3 – 31x2 – 26x + 24 = 0
(I'd started with a degree-five polynomial. Since I've effectively divided out the factor x + 1, I've reduced the degree of the polynomial by 1. That's how I know the last line of the division represents a degree-four polynomial.)
hope this helps
:)