Write the polynomial equation of least degree that has the roots: -3i, 3i, i, and -i.
Answer:
(x+3i)(x-3i)(x-i)(x+i)=x^4+x^3-x^3i+9x^2+9x-9xi-9i
2)Determine if the expression 4m^5-6m^8+m+3 is a polynomial in one variable. If so, state the degree.
Answer: It is a polynomial in one variable; Degree = 8
Thanks
your first one would be
(x+3i)(x-3i)(x-i)(x+i) which is what you had, but then
=(x^2 - 9i^2)(x^2 - i^2) remember i^2 = -1
= (x^2 + 9)(x^2 + 1)
= .... you can finish that
your second question is correct
You're welcome! If you have any more questions, feel free to ask.
To find the polynomial equation of least degree with the given roots, we can start by using the concept of complex conjugates.
The roots -3i and 3i are conjugates of each other, and so are i and -i. When we multiply conjugates, we get real numbers. In this case, if we multiply (-3i) and (3i), we get (-3i)(3i) = -9i^2 = 9. Similarly, multiplying (i) and (-i) gives us (-i)(i) = -i^2 = 1.
So, the polynomial equation with these roots can be written as:
(x - 3i)(x + 3i)(x - i)(x + i) = (x^2 + 9)(x^2 + 1)
To simplify further, we can multiply the two binomials:
(x^2 + 9)(x^2 + 1) = x^2(x^2 + 1) + 9(x^2 + 1)
= x^4 + x^2 + 9x^2 + 9
= x^4 + 10x^2 + 9
Thus, the polynomial equation of least degree with the given roots is x^4 + 10x^2 + 9.
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To determine if the expression 4m^5 - 6m^8 + m + 3 is a polynomial in one variable, we need to check if the exponents on the variable m are non-negative integers.
In the given expression, we have m raised to the powers of 5 and 8. Since both exponents are non-negative integers, the expression is a polynomial in one variable.
To find the degree of the polynomial, we consider the term with the highest power of m. In this case, it is the term -6m^8. The degree of the polynomial is equal to the exponent of this term, which is 8.
Therefore, the expression 4m^5 - 6m^8 + m + 3 is a polynomial in one variable with a degree of 8.