Chapter 12

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) \(x^{2}-16\) (b) \(x^{2}+16\)

According to the fundamental theorem of algebra, which of the equations in Exercises 1 and 2 have at least one root? (a) \(x^{5}-14 x^{4}+8 x+53=0\) (b) \(4.17 x^{3}+2.06 x^{2}+0.01 x+1.23=0\) (c) \(i x^{2}+(2+3 i) x-17=0\) (d) \(x^{21}+3 x^{0.3}+1=0\)

An equation is given, followed by one or more roots of the equation. In each case, determine the remaining roots. $$x^{2}-14 x+53=0 ; x=7-2 i$$

(a) State the rational roots theorem. (b) List the possibilities for the rational roots of the equation \(x^{7}-144 x^{2}-8 x-11=0\)

Determine whether the given value for the variable is a root of the equation. $$12 x-8=112 ; x=10$$

In any computation involving complex numbers, express your answer in the form \(a+b i,\) where a and b are real numbers. If \(a\) or \(b,\) or both are zero, then simplify further. Complete the table. $$\begin{array}{llllll} i^{2} & i^{3} & i^{4} & i^{5} & i^{6} & i^{7} & i^{8} \\ \hline-1 & & & & & \\ \hline \end{array}$$

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) \(x^{2}+17\) (b) \(x^{2}-17\)

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

Use the rational roots theorem to list the possibilities for the rational roots of each equation. (a) \(x^{4}-32 x^{3}+40 x^{2}+12 x-3=0\) (b) \(3 x^{4}-32 x^{3}+40 x^{2}+12 x-1=0\)

Determine whether the given value for the variable is a root of the equation. $$12 x^{2}-x-20=0 ; x=5 / 4$$

In any computation involving complex numbers, express your answer in the form \(a+b i,\) where a and b are real numbers. If \(a\) or \(b,\) or both are zero, then simplify further. Simplify the following expression, and write the answer in the form \(a+b i\) $$ 1+3 i-5 i^{2}+4-2 i-i^{3} $$

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. $$x^{2}-3 x+2.26=0$$

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 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) \(16 i\)

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) \(x^{2}+24 x+144\)

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. $$x^{2}-2 x-290=0$$

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

Determine whether the given value for the variable is a root of the equation. $$1-x+x^{2}-x^{3}=0 ; x=-1$$

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. $$\frac{3 x^{2}+4 x-1}{x-1}$$

(a) factor the denominator of the given \(\mathrm{ra}\) tional expression; (b) determine the form of the partial fraction decomposition for the given rational expression; and (c) determine the values of the constants in the partial fraction decomposition that you gave in part (b). To help you in spotting errors, use the fact that in part (c), each of the required constants turns out to be an integer. $$\frac{11 x+30}{x^{2}-100}$$

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. $$x^{3}-3 x^{2}+3=0$$

List the possibilities for rational roots. $$4 x^{3}-9 x^{2}-15 x+3=0$$

An equation is given, followed by one or more roots of the equation. In each case, determine the remaining roots. $$x^{4}+10 x^{3}+38 x^{2}+66 x+45=0 ; x=-2+i$$

Determine whether the given value for the variable is a root of the equation. $$2 x^{2}-3 x+1=0 ; x=1 / 2$$

Determine the real numbers \(c\) and \(d\) such that $$ 8-3 i=2 c+d i $$

(a) factor the denominator of the given \(\mathrm{ra}\) tional expression; (b) determine the form of the partial fraction decomposition for the given rational expression; and (c) determine the values of the constants in the partial fraction decomposition that you gave in part (b). To help you in spotting errors, use the fact that in part (c), each of the required constants turns out to be an integer. $$\frac{x+18}{x^{2}-36}$$

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. $$x^{4}-3 x^{2}+3=0$$

List the possibilities for rational roots. $$x^{4}-x^{3}+10 x^{2}-24=0$$

Determine whether the given value for the variable is a root of the equation. $$(x-1)(x-2)(x-3)=0 ; x=4$$

(a) factor the denominator of the given \(\mathrm{ra}\) tional expression; (b) determine the form of the partial fraction decomposition for the given rational expression; and (c) determine the values of the constants in the partial fraction decomposition that you gave in part (b). To help you in spotting errors, use the fact that in part (c), each of the required constants turns out to be an integer. $$\frac{8 x-2 \sqrt{5}}{x^{2}-5}$$

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. $$x^{4}+x^{3}+x^{2}+x+1=0$$

List the possibilities for rational roots. $$8 x^{5}-x^{2}+9=0$$

An equation is given, followed by one or more roots of the equation. In each case, determine the remaining roots. $$4 x^{3}-47 x^{2}+232 x+61=0 ; x=6-5 i$$

Determine whether the given value is a zero of the function. $$f(x)=3 x-2 ; x=2 / 3$$

Simplify each of the following. (a) \((5-6 i)+(9+2 i)\) (b) \((5-6 i)-(9+2 i)\)

(a) factor the denominator of the given \(\mathrm{ra}\) tional expression; (b) determine the form of the partial fraction decomposition for the given rational expression; and (c) determine the values of the constants in the partial fraction decomposition that you gave in part (b). To help you in spotting errors, use the fact that in part (c), each of the required constants turns out to be an integer. $$\frac{2 \sqrt{11}}{x^{2}-11}$$

Determine the constants (denoted by capital letters) so that each equation is an identity. For Exercises 1– 6, do each problem in two ways: (a) use the equating-the-coefficients theorem, as in Example 1; and (b) use the convenient-values method that was explained after Example 1. For the remainder of the exercises, use either method (or a combination). $$\frac{7 x}{(x-5)^{2}}=\frac{A}{x-5}+\frac{B}{(x-5)^{2}}$$

List the possibilities for rational roots. $$18 x^{4}-10 x^{3}+x^{2}-4=0$$

An equation is given, followed by one or more roots of the equation. In each case, determine the remaining roots. $$9 x^{4}+18 x^{3}+20 x^{2}-32 x-64=0 ; x=-1+\sqrt{3} i$$

Determine whether the given value is a zero of the function. $$g(x)=1+x^{2} ; x=-1$$

If \(z=1+4 i,\) compute \(z-10 i\)

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. $$\frac{4 x^{3}-x^{2}+8 x-1}{x^{2}-x+1}$$

(a) factor the denominator of the given \(\mathrm{ra}\) tional expression; (b) determine the form of the partial fraction decomposition for the given rational expression; and (c) determine the values of the constants in the partial fraction decomposition that you gave in part (b). To help you in spotting errors, use the fact that in part (c), each of the required constants turns out to be an integer. $$\frac{7 x+39}{x^{2}-x-6}$$

List the possibilities for rational roots. $$\frac{2}{3} x^{3}-x^{2}-5 x+2=0$$

An equation is given, followed by one or more roots of the equation. In each case, determine the remaining roots. $$4 x^{4}-32 x^{3}+81 x^{2}-72 x+162=0 ; x=4+\sqrt{2} i$$

Determine whether the given value is a zero of the function. $$h(x)=5 x^{3}-x^{2}+2 x+8 ; x=-1$$

Compute each of the following. (a) \((3-4 i)(5+i)\) (b) \((5+i)(3-4 i)\) (c) \(\frac{3-4 i}{5+i}\) (d) \(\frac{5+i}{3-4 i}\)