Find a polynomial of degree 4 with i as a zero of multiplicity 1, – i as a zero of multiplicity 1, and 5 as a zero of multiplicity 2.
A. x^4 – 10x^3 + 26x^2 – 10x + 25
B. x^4 – 10x^3 + 24x^2 + 10x – 25
C. x^4 – 24x^2 + 25
D. x^4 – 26x^2 – 25
i say the answer is C... Any thoughts? Thanks!
The quartic polynomial you are looking for has the form kΠ(x-ri) where the summation is over the order of the polynomial (4 in the given case) and ri is the ith zero. k is a constant of multiplication, and is reflected in the coefficient of the highest order term.
Among the four choices, the coefficient of the 4th degree term is 1, so k=1.
From the given roots, we conclude that the polynomial is
(x-i)(x+i)(x-5)²
You only have to expand the product to find out which is the correct choice.
Note that the first two factors expand to (x²-i²)=(x²+1).
Interesting explanation but is my answer correct? I am not too sure? I am still saying C. x^4 – 24x^2 + 25
Let's look at the expression
(x²+1)(x-5)²
=(x²+1)(x²-10x+25)
The constant term is +25, so it can only be either A or C.
To see which one we should choose, we would calculate the coefficient of the terms which differ between the two choices.
Can you calculate the coefficient of the term in x? If it is -10, then the choice is A, if it is 0, the choice is C. (If it is anything else, we're in trouble!!) You should be able to find out by inspection of the simplified expression I gave above.
You can also calculate the x³ term for a check.
Post the result for verification if you wish.
Based on your explanation I am goin with C am I correct?
C. x^4 – 24x^2 + 25
If you calculate the coefficient of x, you will find only one combination of terms that give x, namely 1*(-10x) = -10x.
Again, there is only one term combination that gives x³, namely x²*(-10x).
So the coefficients of both x and x³ are -10. Which is going to be your choice?
3x+4=25
To find the polynomial of degree 4 with the given zeros, we can use the fact that if a complex number (such as i or -i) is a zero of a polynomial with real coefficients, then its conjugate is also a zero of that polynomial.
Therefore, since i is a zero of multiplicity 1, -i is also a zero of multiplicity 1. Similarly, since 5 is a zero of multiplicity 2, we know 5 is also a zero of multiplicity 2.
To construct the polynomial, we can start by writing the factors corresponding to each zero:
(x - i)(x + i) - for the zeros i and -i
(x - 5)(x - 5) - for the zero 5 with multiplicity 2
Now, we can multiply these factors together to find the polynomial:
(x - i)(x + i)(x - 5)(x - 5)
Expanding this expression, we get:
(x^2 - i^2)(x - 5)^2
Since i^2 = -1, we can simplify further:
(x^2 + 1)(x - 5)^2
Expanding again:
(x^2 + 1)(x^2 - 10x + 25)
Now, multiplying these factors, we get:
x^4 - 10x^3 + 25x^2 + x^2 - 10x + 25
Combining like terms, we have:
x^4 - 10x^3 + 26x^2 - 10x + 25
Therefore, the correct polynomial is A) x^4 – 10x^3 + 26x^2 – 10x + 25.
So, the answer you provided, C, is incorrect.