4.4 Polynomial equations (in class) – Conjugate and Descartes

Let [math]f(x)[/math] be a real polynomial with zeros [math]3+\sqrt{2}i, -4[/math], and [math]0[/math]. If the leading coefficient of [math]f(x)[/math] is 1, what is [math]f(1)[/math]?

If [math]f(x)[/math] is a real polyomial with zeros [math]4-i[/math] and [math]-2[/math], which of the following could be [math]f(x)[/math]?

If [math]x=2-i[/math] is a solution to the equation [math]x^4-x^3-6x^2+11x+5=0[/math], which of the following numbers is/are also the solution(s) to the equation?

Let [math]f(x)[/math] be a real polynomial function. If equation [math]f(x)=0[/math] has no real solutions, which of the following functions must have at least one real solutions?

Let [math]f(x)[/math] be a real polynomial function with [math]x[/math]-intercepts [math]2[/math] and [math]-3[/math]. The range of [math]f(x)[/math] is [math](-\infty,0][/math]. If [math]4+2i[/math] is a zero of [math]f(x)[/math], what is [math]f(x)[/math]? Do not expand.

If [math]3+i[/math] is a solution to the real polynomial equation [math]x^4-2x^3+ax^2+bx+10=0[/math]

(a) what is the value of [math]a[/math]?

(b) what is the value of [math]b[/math]?

(c) how many nonreal solutions does the equation have?

Consider a polynomial function [math]f(x)=12x^4-7x^3+13x^2-7x+1[/math].

(a) According to the Descartes’ rule, what is the maximum number of negative zeros of [math]f(x)[/math]?

(b) By (a), how many possible linear factor does [math]f(x)[/math] have?

(c) Given that [math]x=\dfrac{1}{4}[/math] is a zero of [math]f(x)[/math], find the other real solution(s). If you have two or more answers, write the biggest one.

If [math]f(x)[/math] is a real polyomial with zeros [math]2-\sqrt{2}[/math] and [math]2+i[/math], which of the following is/are the factor(s) of [math]f(x)[/math]?

What are the possible number combinations of real zeros of the function [math]f(x)=5x^4-3x^3+2x-1[/math]?

If two of the zeros of the polynomial [math]f(x)=x^4+ax^3+bx^2+cx+d[/math], which has rational coefficients, are [math]-2i[/math] and [math]2-\sqrt{3}[/math], find the value of [math]a+b+c+d[/math].

Consider the polynomial equation of least degree with coefficients that are relatively prime integers. The known roots of this equation are [math]2+i[/math], [math]-\frac{3}{2}[/math] and [math]i[/math]. Find the coefficient of the [math]x^2[/math] term of this equation in expanded form.