Answer:
[tex]\textsf{5)}\quad f(x)=x^3+x^2+4x+4[/tex]
[tex]\textsf{3)} \quad g(x)=x^3-4x^2-11x+30[/tex]
[tex]\textsf{4)} \quad h(x)=x^2-(6+\sqrt{2})x+3\sqrt{2}+9[/tex]
Step-by-step explanation:
Question 5
According to the Complex Conjugate Root Theorem, if -2i is a zero of the polynomial, then its conjugate 2i is also a zero. So, the zeros of the polynomial are -2i, 2i and -1.
The Factor Theorem states that if x = c is a zero of a polynomial, then (x - c) is a factor of that polynomial. Therefore, the factors of the polynomial are:
[tex](x + 2i)[/tex]
[tex](x - 2i)[/tex]
[tex](x + 1)[/tex]
So, the polynomial in factored form is:
[tex]f(x)=(x+2i)(x-2i)(x+1)[/tex]
To write the polynomial in standard form, we can expand the brackets, remembering that [tex]i^2=-1[/tex]:
[tex]f(x)=(x+2i)(x-2i)(x+1)[/tex]
[tex]f(x)=(x^2-2ix+2ix-4i^2)(x+1)[/tex]
[tex]f(x)=(x^2-4i^2)(x+1)[/tex]
[tex]f(x)=(x^2-4(-1))(x+1)[/tex]
[tex]f(x)=(x^2+4)(x+1)[/tex]
[tex]f(x)=x^3+x^2+4x+4[/tex]
[tex]\hrulefill[/tex]
Question 3
The zeros of the polynomial are 2, -3 and 5.
The Factor Theorem states that if x = c is a zero of a polynomial, then (x - c) is a factor of that polynomial. Therefore, the factors of the polynomial are:
[tex](x-2)[/tex]
[tex](x +3)[/tex]
[tex](x -5)[/tex]
So, the polynomial in factored form is:
[tex]g(x)=(x-2)(x+3)(x-5)[/tex]
To write the polynomial in standard form, we can expand the brackets:
[tex]g(x)=(x-2)(x+3)(x-5)[/tex]
[tex]g(x)=(x^2+3x-2x-6)(x-5)[/tex]
[tex]g(x)=(x^2+x-6)(x-5)[/tex]
[tex]g(x)=x^3-5x^2+x^2-5x-6x+30[/tex]
[tex]g(x)=x^3-4x^2-11x+30[/tex]
[tex]\hrulefill[/tex]
Question 4
The zeros of the polynomial are 3 and 3 + √2.
The Factor Theorem states that if x = c is a zero of a polynomial, then (x - c) is a factor of that polynomial. Therefore, the factors of the polynomial are:
[tex](x-3)[/tex]
[tex](x -3-\sqrt{2})[/tex]
So, the polynomial in factored form is:
[tex]h(x)=(x-3)(x -3-\sqrt{2})[/tex]
To write the polynomial in standard form, we can expand the brackets:
[tex]h(x)=(x-3)(x -3-\sqrt{2})[/tex]
[tex]h(x)=x^2-3x-\sqrt{2}x-3x+9+3\sqrt{2}[/tex]
[tex]h(x)=x^2-(6+\sqrt{2})x+3\sqrt{2}+9[/tex]
