91Ó°ÊÓ

An equation is given, followed by one or more roots of the equation. In each case, determine the remaining roots. $$x^{2}-x-\frac{1535}{4}=0 ; x=\frac{1}{2}+8 \sqrt{6}$$

Determine whether the given quadratic polynomial is irreducible. [Recall from the text that a quadratic polynomial \(f(x)\) is irreducible if the equation \(f(x)=0\) has no real roots] (a) \(x^{2}+3 x-4\) (b) \(x^{2}+3 x+4\)

You are given a polynomial equation \(f(x)=0 .\) According to the fundamental theorem of algebra each of these equations has at least one root. However, the fundamental theorem does not tell you whether the equation has any real-number roots. Use a graph to determine whether the equation has at least one real root. Note: You are not being asked to solve the equation. $$x^{2}-3 x+2.26=0$$

Determine whether the given value for the variable is a root of the equation. $$x^{2}-2 x-4=0 ; x=1-\sqrt{5}$$

For Exercises specify the real and imaginary parts of each complex number. (a) \(4+5 i\) (b) \(4-5 i\) (c) \(\frac{1}{2}-i\) (d) \(16 i\)

An equation is given, followed by one or more roots of the equation. In each case, determine the remaining roots. $$x^{3}-13 x^{2}+59 x-87=0 ; x=5+2 i$$

Determine whether the given quadratic polynomial is irreducible. [Recall from the text that a quadratic polynomial \(f(x)\) is irreducible if the equation \(f(x)=0\) has no real roots] (a) \(24 x^{2}+x-3\) (b) \(x^{2}+24 x+144\)

An equation is given, followed by one or more roots of the equation. In each case, determine the remaining roots. $$x^{4}-10 x^{3}+30 x^{2}-10 x-51=0 ; x=4+i$$

For Exercises specify the real and imaginary parts of each complex number. (a) \(-2+\sqrt{7} i\) (b) \(1+5^{1 / 3} i\) (c) \(-3 i\) (d) 0

Use long division to find the quotients and the remainders. Also, write each answer in the form \(p(x)=d(x) \cdot q(x)+R(x),\) as in equation (2) in the text. $$\frac{3 x^{2}+4 x-1}{x-1}$$