Section 1
Division of Polynomials
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.$$\frac{x^{2}-8 x+4}{x-3}$$
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.$$\frac{x^{3}-4 x^{2}+x-2}{x-5}$$
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.$$\frac{x^{2}-6 x-2}{x+5}$$
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.$$\frac{3 x^{2}+4 x-1}{x-1}$$
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.$$\frac{6 x^{3}-2 x+3}{2 x+1}$$
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.$$\frac{x^{4}-4 x^{3}+6 x^{2}-4 x+1}{x-1}$$
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.$$\frac{x^{5}+2}{x+3}$$
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.$$\frac{4 x^{3}-x^{2}+8 x-1}{x^{2}-x+1}$$
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.$$\frac{x^{6}-64}{x-2}$$
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.$$\frac{x^{6}+64}{x-2}$$
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.$$\frac{5 x^{4}-3 x^{2}+2}{x^{2}-3 x+5}$$
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.$$\frac{8 x^{6}-36 x^{4}+54 x^{2}-27}{2 x^{2}-3}$$
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.$$\frac{3 y^{3}-4 y^{2}-3}{y^{2}+5 y+2}$$
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.$$\frac{4 y^{4}-y^{3}+2 y-1}{2 y^{2}-3 y-4}$$
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.$$\frac{t^{4}-4 t^{3}+4 t^{2}-16}{t^{2}-2 t+4}$$
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.$$\frac{2 f^{5}-6 t^{4}-t^{2}+2 t+3}{t^{3}-2}$$
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.$$\frac{z^{5}-1}{z-1}$$
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.$$\frac{1+z+z^{2}+z^{3}}{1+z+z^{2}}$$
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.$$\frac{a x^{2}+b x+c}{x-r}$$
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.$$\frac{a x^{3}+b x^{2}+c x+d}{x-r}$$
In Exercises $21-40$, use synthetic division to find the quotients and remainders. Also, in each case, write the result of the division in the form $p(x)=d(x) \cdot q(x)+R(x),$ as in equation ( 2 ) in the text.$$\frac{x^{2}-6 x-2}{x-5}$$
Use synthetic division to find the quotients and remainders. Also, in each case, write the result of the division in the form $p(x)=d(x) \cdot q(x)+R(x),$ as in equation ( 2 ) in the text.$$\frac{3 x^{2}+4 x-1}{x-1}$$
Use synthetic division to find the quotients and remainders. Also, in each case, write the result of the division in the form $p(x)=d(x) \cdot q(x)+R(x),$ as in equation ( 2 ) in the text.$$\frac{4 x^{2}-x-5}{x+1}$$
Use synthetic division to find the quotients and remainders. Also, in each case, write the result of the division in the form $p(x)=d(x) \cdot q(x)+R(x),$ as in equation ( 2 ) in the text.$$\frac{x^{2}-1}{x+2}$$
Use synthetic division to find the quotients and remainders. Also, in each case, write the result of the division in the form $p(x)=d(x) \cdot q(x)+R(x),$ as in equation ( 2 ) in the text.$$\frac{6 x^{3}-5 x^{2}+2 x+1}{x-4}$$
Use synthetic division to find the quotients and remainders. Also, in each case, write the result of the division in the form $p(x)=d(x) \cdot q(x)+R(x),$ as in equation ( 2 ) in the text.$$\frac{x^{4}-4 x^{3}+6 x^{2}-4 x+1}{x-1}$$
Use synthetic division to find the quotients and remainders. Also, in each case, write the result of the division in the form $p(x)=d(x) \cdot q(x)+R(x),$ as in equation ( 2 ) in the text.$$\frac{x^{3}-1}{x-2}$$
Use synthetic division to find the quotients and remainders. Also, in each case, write the result of the division in the form $p(x)=d(x) \cdot q(x)+R(x),$ as in equation ( 2 ) in the text.$$\frac{x^{3}-8}{x-2}$$
Use synthetic division to find the quotients and remainders. Also, in each case, write the result of the division in the form $p(x)=d(x) \cdot q(x)+R(x),$ as in equation ( 2 ) in the text.$$\frac{x^{5}-1}{x+2}$$
Use synthetic division to find the quotients and remainders. Also, in each case, write the result of the division in the form $p(x)=d(x) \cdot q(x)+R(x),$ as in equation ( 2 ) in the text.$$\frac{x^{3}-8 x^{2}-1}{x+3}$$
Use synthetic division to find the quotients and remainders. Also, in each case, write the result of the division in the form $p(x)=d(x) \cdot q(x)+R(x),$ as in equation ( 2 ) in the text.$$\frac{x^{4}-6 x^{3}+2}{x+4}$$
Use synthetic division to find the quotients and remainders. Also, in each case, write the result of the division in the form $p(x)=d(x) \cdot q(x)+R(x),$ as in equation ( 2 ) in the text.$$\frac{3 x^{3}-2 x^{2}+x+1}{x-\frac{1}{2}}$$
Use synthetic division to find the quotients and remainders. Also, in each case, write the result of the division in the form $p(x)=d(x) \cdot q(x)+R(x),$ as in equation ( 2 ) in the text.$$\frac{x^{3}-4 x^{2}-3 x+6}{x-10}$$
Use synthetic division to find the quotients and remainders. Also, in each case, write the result of the division in the form $p(x)=d(x) \cdot q(x)+R(x),$ as in equation ( 2 ) in the text.$$\frac{1+3 x+3 x^{2}+x^{3}}{x+1}$$
Use synthetic division to find the quotients and remainders. Also, in each case, write the result of the division in the form $p(x)=d(x) \cdot q(x)+R(x),$ as in equation ( 2 ) in the text.$$\frac{x^{3}-x^{2}}{x+5}$$
Use synthetic division to find the quotients and remainders. Also, in each case, write the result of the division in the form $p(x)=d(x) \cdot q(x)+R(x),$ as in equation ( 2 ) in the text.$$\frac{5 x^{4}-4 x^{3}+3 x^{2}-2 x+1}{x+\frac{1}{2}}$$
Use synthetic division to find the quotients and remainders. Also, in each case, write the result of the division in the form $p(x)=d(x) \cdot q(x)+R(x),$ as in equation ( 2 ) in the text.$$\frac{14-27 x-27 x^{2}+54 x^{3}}{x-\frac{2}{3}}$$
Use synthetic division to find the quotients and remainders. Also, in each case, write the result of the division in the form $p(x)=d(x) \cdot q(x)+R(x),$ as in equation ( 2 ) in the text.$$\frac{14-27 x-27 x^{2}+54 x^{3}}{x+\frac{2}{3}}$$
Use synthetic division to find the quotients and remainders. Also, in each case, write the result of the division in the form $p(x)=d(x) \cdot q(x)+R(x),$ as in equation ( 2 ) in the text.$$\frac{x^{4}+3 x^{2}+12}{x-3}$$
Use synthetic division to find the quotients and remainders. Also, in each case, write the result of the division in the form $p(x)=d(x) \cdot q(x)+R(x),$ as in equation ( 2 ) in the text.(a) $\frac{x^{4}-16}{x-2}$(b) $\frac{x^{4}+16}{x+2}$
Each expression has the form $x^{n}-a^{n}$ Write each expression as a product of two factors (as in the box on page 924 ).$$x^{5}-32$$
Each expression has the form $x^{n}-a^{n}$ Write each expression as a product of two factors (as in the box on page 924 ).$$y^{6}-1$$
Each expression has the form $x^{n}-a^{n}$ Write each expression as a product of two factors (as in the box on page 924 ).$$z^{4}-81$$
Each expression has the form $x^{n}-a^{n}$ Write each expression as a product of two factors (as in the box on page 924 ).$$x^{7}-y^{7}$$
Use synthetic division to determine the quotient $q(x)$ and the remainder $R(x)$ in each case.$$\frac{6 x^{2}-8 x+1}{3 x-4}$$Hint: Divide both numerator and denominator by $3 .$ (Why?)
Use synthetic division to determine the quotient $q(x)$ and the remainder $R(x)$ in each case.$$\frac{4 x^{3}+6 x^{2}-6 x-5}{2 x-3}$$
Use synthetic division to determine the quotient $q(x)$ and the remainder $R(x)$ in each case.$$\frac{6 x^{3}+1}{2 x+1}$$
Use synthetic division to determine the quotient $q(x)$ and the remainder $R(x)$ in each case.$$\frac{5 x^{3}-3 x^{2}+1}{3 x+1}$$
When $x^{3}+k x+1$ is divided by $x+1,$ the remainder is $-4 .$ Find $k$
(a) Show that when $x^{3}+k x+6$ is divided by $x+3,$ the remainder is $-21-3 k$(b) Determine a value of $k$ such that $x+3$ will be a factor of $x^{3}+k x+6$
When $x^{2}+2 p x-3 q^{2}$ is divided by $x-p$, the remainder is zero. Show that $p^{2}=q^{2}$
Given that $x-3$ is a factor of $x^{3}-2 x^{2}-4 x+3,$ solve the equation $x^{3}-2 x^{2}-4 x+3=0$
The process of synthetic division applies equally well when some or all of the coefficients are nonreal complex numbers. In Exercises $53-56$, use synthetic division to determine the quotient $q(x)$ and the remainder $R(x)$ in each case.$$\frac{x^{2}-4 x+1}{x-i}$$
The process of synthetic division applies equally well when some or all of the coefficients are nonreal complex numbers. Use synthetic division to determine the quotient $q(x)$ and the remainder $R(x)$ in each case.$$\frac{x^{3}-2 x^{2}-4}{x-3 i}$$
The process of synthetic division applies equally well when some or all of the coefficients are nonreal complex numbers. Use synthetic division to determine the quotient $q(x)$ and the remainder $R(x)$ in each case.$$\frac{x^{2}-2 x+2}{x-(1+i)}$$
The process of synthetic division applies equally well when some or all of the coefficients are nonreal complex numbers. Use synthetic division to determine the quotient $q(x)$ and the remainder $R(x)$ in each case.$$\frac{x^{3}-x^{2}+4 x-4}{x+2 i}$$
Given that the identity $f(x)=d(x) \cdot q(x)+R(x)$ holds for the following polynomials, evaluate $f(\sqrt{3})$ Hint (of sorts): There's an easy way and a tedious way.$$\begin{array}{ll}f(t)=\ell-3 t^{4}+2 t^{3}-5 t^{2}+6 t-7 & d(t)=t-4 \\q(t)=t^{4}+t^{3}+6 t^{2}+19 t+82 & R(t)=321\end{array}$$
Given that the identity $f(t)=d(t) \cdot q(t)+R(t)$ holds for the following polynomials, evaluate $f(4)$$$\begin{aligned}&f(t)=t^{5}-3 t^{4}+2 t^{3}-5 t^{2}+6 t-7 \quad d(t)=t-4\\&q(t)=t^{4}+t^{3}+6 t^{2}+19 t+82 \quad R(t)=321\end{aligned}$$
Find the remainder when $t^{5}-5 a^{4} t+4 a^{5}$ is divided by $t-a$
When $f(x)$ is divided by $(x-a)(x-b),$ the remainder is $A x+B .$ Apply the division algorithm to show that$$A=\frac{f(a)-f(b)}{a-b} \quad \text { and } \quad B=\frac{b f(a)-a f(b)}{b-a}$$