Consider the polynomial f(x) = 5x4+ 3x2+ x + 19. How many complex solutions does the polynomial have?
1 answer:
Answer:
- there are 4 complex solutions
- 3 real zeros and 2 complex zeros
Step-by-step explanation:
1. Descarte's rule of signs tells you there are 0 positive real roots and 0 or 2 negative real roots. (for positive x, signs are ++++ so have no changes; for negative x, signs are ++-+, so have 2 changes.) A graph shows no real roots.
2. There are 3 sign changes in the given polynomial, so 3 or 1 positive real roots. When the sign of x is changed, there are 2 sign changes, so 0 or 2 negative real roots. A graph shows 2 negative and one positive real root (for a total of 3), so the remaining 2 roots are complex.
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Given:
The expression is:
To find:
Part A: The expression using parentheses so that the expression equals 23.
Part B: The expression using parentheses so that the expression equals 3.
Solution:
Part A:
In option A,
[Using BODMAS]
In option B,
[Using BODMAS]
In option C,
In option D,
[Using BODMAS]
After the calculation, we have and
.
Therefore, the correct options are B and D.
Part B: From part A, it is clear that
Therefore, the correct option is C.
For this case we must simplify the following expression:
We eliminate the parentheses taking into account that:
We add similar terms taking into account that:
Equal signs are added and the same sign is placed.
Different signs are subtracted and the major sign is placed.
Answer: