Form a third degree polynomial function with real coefficients such that 2+i and -5 are zeros.
f(x)= ?
since we want real coefficients, complex roots must come in conjugate pairs. So,
f(x) = (x+5)(x-(2+i))(x-(2-i))
= x^3 + x^2 - 15x + 25
To form a third-degree polynomial function with real coefficients given that 2+i and -5 are zeros, we can use the concept of complex conjugate pairs. Since the coefficients are real, the complex conjugate of 2+i, which is 2-i, will also be a zero of the polynomial.
To find the polynomial, we can use the factored form of a polynomial:
f(x) = (x - x1)(x - x2)(x - x3)
where x1, x2, and x3 are the zeros of the polynomial.
Given that 2+i and its conjugate 2-i are two zeros of the polynomial, along with -5, we can write the polynomial as:
f(x) = (x - (2+i))(x - (2-i))(x - (-5))
Expanding this expression, we have:
f(x) = (x - 2 - i)(x - 2 + i)(x + 5)
Multiplying the first two factors using the difference of squares formula (a^2 - b^2 = (a + b)(a - b)):
f(x) = ((x - 2)^2 - i^2)(x + 5)
f(x) = ((x - 2)^2 + 1)(x + 5)
Expanding further:
f(x) = (x^2 - 4x + 4 + 1)(x + 5)
f(x) = (x^2 - 4x + 5)(x + 5)
Finally, expanding the last set of parentheses:
f(x) = x^3 + x^2 - 19x + 25
Therefore, the third-degree polynomial function with real coefficients that has 2+i and -5 as zeros is:
f(x) = x^3 + x^2 - 19x + 25
To form a third degree polynomial with real coefficients given the zeros 2+i and -5, we know that complex zeros always come in conjugate pairs. Therefore, the conjugate of 2+i is 2-i.
To find the polynomial, we can start with the factored form:
f(x) = (x - zero1)(x - zero2)(x - zero3)
Substituting the given zeros, we have:
f(x) = (x - 2-i)(x - 2+i)(x - (-5))
Using the difference of squares formula (a^2 - b^2 = (a+b)(a-b)), we can simplify the expression further:
f(x) = [(x - 2)^2 - (i)^2](x + 5)
f(x) = [(x - 2)^2 - (-1)](x + 5)
f(x) = [(x - 2)^2 + 1](x + 5)
f(x) = (x^2 - 4x + 4 + 1)(x + 5)
f(x) = (x^2 - 4x + 5)(x + 5)
Expanding further, we get:
f(x) = x^3 + x^2 - 19x - 25
Therefore, the third degree polynomial with real coefficients that has the zeros 2+i and -5 is f(x) = x^3 + x^2 - 19x - 25.