Search

Text Color

Margin Size

Font Type

Enable Dyslexic Font

25.3: Exercises

Last updated

May 2, 2022
Save as PDF
- 25.2: Binomial Expansion
- 26: Appendix A - Introduction to the TI-84

Thomas Tradler and Holly Carley
CUNY New York City College of Technology via New York City College of Technology at CUNY Academic Works

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\id}{\mathrm{id}}$ $\newcommand{\Span}{\mathrm{span}}$

( \newcommand{\kernel}{\mathrm{null}\,}\) $\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$ $\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$ $\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\id}{\mathrm{id}}$

$\newcommand{\Span}{\mathrm{span}}$

$\newcommand{\kernel}{\mathrm{null}\,}$

$\newcommand{\range}{\mathrm{range}\,}$

$\newcommand{\RealPart}{\mathrm{Re}}$

$\newcommand{\ImaginaryPart}{\mathrm{Im}}$

$\newcommand{\Argument}{\mathrm{Arg}}$

$\newcommand{\norm}[1]{\| #1 \|}$

$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$

$\newcommand{\Span}{\mathrm{span}}$ $\newcommand{\AA}{\unicode[.8,0]{x212B}}$

$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$

$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$

$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vectorC}[1]{\textbf{#1}}$

$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$

$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$

$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$

$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$

$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$

$\newcommand{\avec}{\mathbf a}$

$\newcommand{\bvec}{\mathbf b}$

$\newcommand{\cvec}{\mathbf c}$

$\newcommand{\dvec}{\mathbf d}$

$\newcommand{\dtil}{\widetilde{\mathbf d}}$

$\newcommand{\evec}{\mathbf e}$

$\newcommand{\fvec}{\mathbf f}$

$\newcommand{\nvec}{\mathbf n}$

$\newcommand{\pvec}{\mathbf p}$

$\newcommand{\qvec}{\mathbf q}$

$\newcommand{\svec}{\mathbf s}$

$\newcommand{\tvec}{\mathbf t}$

$\newcommand{\uvec}{\mathbf u}$

$\newcommand{\vvec}{\mathbf v}$

$\newcommand{\wvec}{\mathbf w}$

$\newcommand{\xvec}{\mathbf x}$

$\newcommand{\yvec}{\mathbf y}$

$\newcommand{\zvec}{\mathbf z}$

$\newcommand{\rvec}{\mathbf r}$

$\newcommand{\mvec}{\mathbf m}$

$\newcommand{\zerovec}{\mathbf 0}$

$\newcommand{\onevec}{\mathbf 1}$

$\newcommand{\real}{\mathbb R}$

$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$

$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$

$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$

$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$

$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$

$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$

$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$

$\newcommand{\bcal}{\cal B}$

$\newcommand{\ccal}{\cal C}$

$\newcommand{\scal}{\cal S}$

$\newcommand{\wcal}{\cal W}$

$\newcommand{\ecal}{\cal E}$

$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$

$\newcommand{\gray}[1]{\color{gray}{#1}}$

$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$

$\newcommand{\rank}{\operatorname{rank}}$

$\newcommand{\row}{\text{Row}}$

$\newcommand{\col}{\text{Col}}$

$\renewcommand{\row}{\text{Row}}$

$\newcommand{\nul}{\text{Nul}}$

$\newcommand{\var}{\text{Var}}$

$\newcommand{\corr}{\text{corr}}$

$\newcommand{\len}[1]{\left|#1\right|}$

$\newcommand{\bbar}{\overline{\bvec}}$

$\newcommand{\bhat}{\widehat{\bvec}}$

$\newcommand{\bperp}{\bvec^\perp}$

$\newcommand{\xhat}{\widehat{\xvec}}$

$\newcommand{\vhat}{\widehat{\vvec}}$

$\newcommand{\uhat}{\widehat{\uvec}}$

$\newcommand{\what}{\widehat{\wvec}}$

$\newcommand{\Sighat}{\widehat{\Sigma}}$

$\newcommand{\lt}{<}$

$\newcommand{\gt}{>}$

$\newcommand{\amp}{&}$

$\definecolor{fillinmathshade}{gray}{0.9}$

Exercise $\PageIndex{1}$

Find the value of the factorial or binomial coefficient.

$5!$
$3!$
$9!$
$2!$
$0!$
$1!$
$19!$
$64!$
$\dbinom{5}{2}$
$\dbinom{9}{6}$
$\dbinom{12}{1}$
$\dbinom{12}{0}$
$\dbinom{23}{22}$
$\dbinom{19}{12}$
$\dbinom{13}{11}$
$\dbinom{16}{5}$

Answer

$120$
$6$
$362880$
$2$
$1$
$1$
$\approx 1.216 \cdot 10^{17}$
$\approx 1.269 \cdot 10^{89}$
$10$
$84$
$12$
$1$
$23$
$50388$
$78$
$4368$

Exercise $\PageIndex{2}$

Expand the expression via the binomial theorem.

$(m+n)^{4}$
$(x+2)^5$
$(x-y)^6$
$(-p-q)^5$

Answer

$m^{4}+4 m^{3} n+6 m^{2} n^{2}+4 m n^{3}+n^{4}$
$x^{5}+10 x^{4}+40 x^{3}+80 x^{2}+80 x+32$
$x^{6}-6 x^{5} y+15 x^{4} y^{2}-20 x^{3} y^{3}+15 x^{2} y^{4}-6 x y^{5}+y^{6}$
$-p^{5}-5 p^{4} q-10 p^{3} q^{2}-10 p^{2} q^{3}-5 p q^{4}-q^{5}$

Exercise $\PageIndex{3}$

Expand the expression.

$(x-2y)^3$
$(x-10)^4$
$(x^2y+y^2)^5$
$(2y^2-5x^4)^4$
$(x+\sqrt{x})^3$
$\left(-2\dfrac{x^2}{y}-\dfrac{y^3}{x}\right)^5$
$(\sqrt{2}-2\sqrt{3})^3$
$(1-i)^3$

Answer

$x^{3}-6 x^{2} y+12 x y^{2}-8 y^{3}$
$x^{4}-40 x^{3}+600 x^{2}-4000 x+10000$
$x^{10} y^{5}+5 x^{8} y^{6}+10 x^{6} y^{7}+10 x^{4} y^{8}+5 x^{2} y^{9}+y^{10}$
$16 y^{8}-160 x^{4} y^{6}+600 x^{8} y^{4}-1000 x^{12} y^{2}+625 x^{16}$
$x^{3}+3 x^{\frac{5}{2}}+3 x^{2}+x^{\frac{3}{2}}$
$-32 \dfrac{x^{10}}{y^{5}}-80 \dfrac{x^{7}}{y}-80 x^{4} y^{3}-40 x y^{7}-10 \dfrac{y^{11}}{x^{2}}-\dfrac{y^{15}}{x^{5}}$
$38 \sqrt{2}-36 \sqrt{3}$
$-2-2 i$

Exercise $\PageIndex{4}$

Determine:

the first $3$ terms in the binomial expansion of $(xy-4x)^{5}$
the first $2$ terms in the binomial expansion of $(2a^2+b^3)^{9}$
the last $3$ terms in the binomial expansion of $(3y^2-x^2)^{7}$
the first $3$ terms in the binomial expansion of $\left(\dfrac{x}{y}-\dfrac{y}{x}\right)^{10}$
the last $4$ terms in the binomial expansion of $\left(m^3n+\dfrac{1}{2}n^2\right)^{6}$

Answer

$x^{5} y^{5}-20 x^{5} y^{4}+160 x^{5} y^{3}$
$512 a^{18}+2304 a^{16} b^{3}$
$-189 x^{10} y^{4}+21 x^{12} y^{2}-x^{14}$
$\dfrac{x^{10}}{y^{10}}-10 \dfrac{x^{8}}{y^{8}}+45 \dfrac{x^{6}}{y^{6}}$
$\dfrac{5}{2} m^{9} n^{9}+\dfrac{15}{16} m^{6} n^{10}+\dfrac{3}{16} m^{3} n^{11}+\dfrac{1}{64} n^{12}$

Exercise $\PageIndex{5}$

Determine:

the $5$ th term in the binomial expansion of $(x+y)^{7}$
the $3$ rd term in the binomial expansion of $(x^2-y)^{9}$
the $10$ th term in the binomial expansion of $(2-w)^{11}$
the $5$ th term in the binomial expansion of $(2x+xy)^{7}$
the $7$ th term in the binomial expansion of $(2a+5b)^{6}$
the $6$ th term in the binomial expansion of $(3p^2-q^3p)^{7}$
the $10$ th term in the binomial expansion of $\left(4+\dfrac{b}{2}\right)^{13}$

Answer

$35 x^{3} y^{4}$
$36 x^{14} y^{2}$
$-220 w^{9}$
$280 x^{7} y^{4}$
$15625 b^{6}$
$-189 p^{9} q^{15}$
$\dfrac{715}{2} b^{9}$

Exercise $\PageIndex{6}$

Determine:

the $x^3y^{6}$ -term in the binomial expansion of $(x+y)^{9}$
the $r^4s^4$ -term in the binomial expansion of $(r^2-s)^{6}$
the $x^{4}$ -term in the binomial expansion of $(x-1)^{11}$
the $x^3y^{6}$ -term in the binomial expansion of $(x^3+5y^2)^{4}$
the $x^{7}$ -term in the binomial expansion of $(2x-x^2)^{5}$
the imaginary part of the number $(1+i)^3$

Answer

$84 x^{3} y^{6}$
$15 r^{4} s^{4}$
$-330 x^{4}$
$500 x^{3} y^{6}$
$80 x^{7}$
$2 i$

Exercise 25.3.1\PageIndex{1}

Exercise 25.3.2\PageIndex{2}

Exercise 25.3.3\PageIndex{3}

Exercise 25.3.4\PageIndex{4}

Exercise 25.3.5\PageIndex{5}

Exercise 25.3.6\PageIndex{6}

Support Center

How can we help?

Exercise $\PageIndex{1}$

Exercise $\PageIndex{2}$

Exercise $\PageIndex{3}$

Exercise $\PageIndex{4}$

Exercise $\PageIndex{5}$

Exercise $\PageIndex{6}$