91Ó°ÊÓ

For Exercises \(119-124,\) factor the expressions over the set of complex numbers. For assistance, consider these examples. \(\cdot\) In Chapter \(\mathrm{R}\) we saw that some expressions factor over the set of integers. For example: \(x^{2}-4=(x+2)(x-2)\). \(\cdot\) Some expressions factor over the set of irrational numbers. For example: \(x^{2}-5=(x+\sqrt{5})(x-\sqrt{5})\). \(\cdot\) To factor an expression such as \(x^{2}+4,\) we need to factor over the set of complex numbers. For example, verify that \(x^{2}+4=(x+2 i)(x-2 i)\). a. \(x^{2}-11\) b. \(x^{2}+11\)

Use a calculator to perform the indicated operations. a. \(\sqrt{-16}\) b. \((4-5 i)-(2+3 i)\) c. \((12-15 i)(-2+9 i)\)

Solve for the indicated variable. . \(x^{2}-x y-2 y^{2}=0\) for \(x\)

Use a calculator to perform the indicated operations. a. \(\sqrt{-169}\) b. \((-11-2 i)+(-4+9 i)\) c. \((8+12 i)(-3-7 i)\)

Write an equation with integer coefficients and the variable \(x\) that has the given solution set. \( \\{4,-2\\}\)

Use a calculator to perform the indicated operations. a. \((11+4 i)^{2}\) b. \(\frac{11}{10 i}\) c. \(\frac{5+7 i}{6+8 i}\)

Write an equation with integer coefficients and the variable \(x\) that has the given solution set. \(\\{7,-1\\}\)

Write an equation with integer coefficients and the variable \(x\) that has the given solution set. \(\\{\sqrt{5},-\sqrt{5}\\}\)

Write an equation with integer coefficients and the variable \(x\) that has the given solution set. \(\\{\sqrt{2},-\sqrt{2}\\}\)

Write an equation with integer coefficients and the variable \(x\) that has the given solution set. \(\\{2 i,-2 i\\}\)