91Ó°ÊÓ

Jordan said that if the roots of a polynomial function \(\mathrm{f}(x)\) are \(r_{1}, r_{2},\) and \(r_{3},\) then the roots of \(\mathrm{g}(x)=\mathrm{f}(x-a)\) are \(r_{1}+a, r_{2}+a,\) and \(r_{3}+a .\) Do you agree with Jordan? Explain why or why not.

Christina said that when \(a, b,\) and \(c\) are rational numbers and \(a\) and \(c\) have opposite signs, the quadratic equation \(a x^{2}+b x+c=0\) must have real roots. Do you agree with Christina? Explain why or why not.

Joshua said that the product of a complex number and its conjugate is always a real number. Do you agree with Joshua? Explain why or why not.

Adrien said that if the roots of a quadratic equation are \(\frac{1}{2}\) and \(\frac{3}{4}\) , the equation is \(4 x^{2}-5 x+\frac{3}{2}=0 .\) Olivia said that the equation is \(8 x^{2}-10 x+3=0 .\) Who is correct? Justify your answer.

Noah said that if \(a, b,\) and \(c\) are rational numbers and \(b^{2}-4 a c<0\) , then the roots of the equation \(a x^{2}+b x+c=0\) are complex conjugates. Do you agree with Noah? Justify your answer.

Ethan said that the square of any pure imaginary number is a negative real number. Do you agree with Ethan? Justify your answer.

Rita said that when \(a, b,\) and \(c\) are real numbers, the roots of \(a x^{2}+b x+c=0\) are real numbers only when \(b^{2} \geq 4 a c .\) Do you agree with Rita? Explain why or why not.

In \(3-18,\) find all roots of each given function by factoring or by using the quadratic formula. $$ \mathrm{f}(x)=x^{3}+7 x^{2}+10 x $$

In \(3-14\) , use the quadratic formula to find the roots of each equation. Irrational roots should be written in simplest radical form. $$ x^{2}+5 x+4=0 $$

Without solving each equation, find the sum and product of the roots. \(x^{2}+x+1=0\)