11.6 Binomial theorem  (Page 3/6)

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Key equations

 Binomial Theorem ${\left(x+y\right)}^{n}=\sum _{k-0}^{n}\left(\begin{array}{c}n\\ k\end{array}\right){x}^{n-k}{y}^{k}$ $\left(r+1\right)th\text{\hspace{0.17em}}$ term of a binomial expansion $\left(\begin{array}{c}n\\ r\end{array}\right){x}^{n-r}{y}^{r}$

Key concepts

• $\left(\begin{array}{c}n\\ r\end{array}\right)\text{\hspace{0.17em}}$ is called a binomial coefficient and is equal to $C\left(n,r\right).\text{\hspace{0.17em}}$ See [link] .
• The Binomial Theorem allows us to expand binomials without multiplying. See [link] .
• We can find a given term of a binomial expansion without fully expanding the binomial. See [link] .

Verbal

What is a binomial coefficient, and how it is calculated?

A binomial coefficient is an alternative way of denoting the combination $\text{\hspace{0.17em}}C\left(n,r\right).\text{\hspace{0.17em}}$ It is defined as $\text{\hspace{0.17em}}\left(\begin{array}{c}n\\ r\end{array}\right)=\text{\hspace{0.17em}}C\left(n,r\right)\text{\hspace{0.17em}}=\frac{n!}{r!\left(n-r\right)!}.$

What role do binomial coefficients play in a binomial expansion? Are they restricted to any type of number?

What is the Binomial Theorem and what is its use?

The Binomial Theorem is defined as $\text{\hspace{0.17em}}{\left(x+y\right)}^{n}=\sum _{k=0}^{n}\left(\begin{array}{c}n\\ k\end{array}\right){x}^{n-k}{y}^{k}\text{\hspace{0.17em}}$ and can be used to expand any binomial.

When is it an advantage to use the Binomial Theorem? Explain.

Algebraic

For the following exercises, evaluate the binomial coefficient.

$\left(\begin{array}{c}6\\ 2\end{array}\right)$

15

$\left(\begin{array}{c}5\\ 3\end{array}\right)$

$\left(\begin{array}{c}7\\ 4\end{array}\right)$

35

$\left(\begin{array}{c}9\\ 7\end{array}\right)$

$\left(\begin{array}{c}10\\ 9\end{array}\right)$

10

$\left(\begin{array}{c}25\\ 11\end{array}\right)$

$\left(\begin{array}{c}17\\ 6\end{array}\right)$

12,376

$\left(\begin{array}{c}200\\ 199\end{array}\right)$

For the following exercises, use the Binomial Theorem to expand each binomial.

${\left(4a-b\right)}^{3}$

$64{a}^{3}-48{a}^{2}b+12a{b}^{2}-{b}^{3}$

${\left(5a+2\right)}^{3}$

${\left(3a+2b\right)}^{3}$

$27{a}^{3}+54{a}^{2}b+36a{b}^{2}+8{b}^{3}$

${\left(2x+3y\right)}^{4}$

${\left(4x+2y\right)}^{5}$

$1024{x}^{5}+2560{x}^{4}y+2560{x}^{3}{y}^{2}+1280{x}^{2}{y}^{3}+320x{y}^{4}+32{y}^{5}$

${\left(3x-2y\right)}^{4}$

${\left(4x-3y\right)}^{5}$

$1024{x}^{5}-3840{x}^{4}y+5760{x}^{3}{y}^{2}-4320{x}^{2}{y}^{3}+1620x{y}^{4}-243{y}^{5}$

${\left(\frac{1}{x}+3y\right)}^{5}$

${\left({x}^{-1}+2{y}^{-1}\right)}^{4}$

$\frac{1}{{x}^{4}}+\frac{8}{{x}^{3}y}+\frac{24}{{x}^{2}{y}^{2}}+\frac{32}{x{y}^{3}}+\frac{16}{{y}^{4}}$

${\left(\sqrt{x}-\sqrt{y}\right)}^{5}$

For the following exercises, use the Binomial Theorem to write the first three terms of each binomial.

${\left(a+b\right)}^{17}$

${a}^{17}+17{a}^{16}b+136{a}^{15}{b}^{2}$

${\left(x-1\right)}^{18}$

${\left(a-2b\right)}^{15}$

${a}^{15}-30{a}^{14}b+420{a}^{13}{b}^{2}$

${\left(x-2y\right)}^{8}$

${\left(3a+b\right)}^{20}$

$3,486,784,401{a}^{20}+23,245,229,340{a}^{19}b+73,609,892,910{a}^{18}{b}^{2}$

${\left(2a+4b\right)}^{7}$

${\left({x}^{3}-\sqrt{y}\right)}^{8}$

${x}^{24}-8{x}^{21}\sqrt{y}+28{x}^{18}y$

For the following exercises, find the indicated term of each binomial without fully expanding the binomial.

The fourth term of $\text{\hspace{0.17em}}{\left(2x-3y\right)}^{4}$

The fourth term of $\text{\hspace{0.17em}}{\left(3x-2y\right)}^{5}$

$-720{x}^{2}{y}^{3}$

The third term of $\text{\hspace{0.17em}}{\left(6x-3y\right)}^{7}$

The eighth term of $\text{\hspace{0.17em}}{\left(7+5y\right)}^{14}$

$220,812,466,875,000{y}^{7}$

The seventh term of $\text{\hspace{0.17em}}{\left(a+b\right)}^{11}$

The fifth term of $\text{\hspace{0.17em}}{\left(x-y\right)}^{7}$

$35{x}^{3}{y}^{4}$

The tenth term of $\text{\hspace{0.17em}}{\left(x-1\right)}^{12}$

The ninth term of $\text{\hspace{0.17em}}{\left(a-3{b}^{2}\right)}^{11}$

$1,082,565{a}^{3}{b}^{16}$

The fourth term of $\text{\hspace{0.17em}}{\left({x}^{3}-\frac{1}{2}\right)}^{10}$

The eighth term of $\text{\hspace{0.17em}}{\left(\frac{y}{2}+\frac{2}{x}\right)}^{9}$

$\frac{1152{y}^{2}}{{x}^{7}}$

Graphical

For the following exercises, use the Binomial Theorem to expand the binomial $f\left(x\right)={\left(x+3\right)}^{4}.$ Then find and graph each indicated sum on one set of axes.

Find and graph $\text{\hspace{0.17em}}{f}_{1}\left(x\right),\text{\hspace{0.17em}}$ such that $\text{\hspace{0.17em}}{f}_{1}\left(x\right)\text{\hspace{0.17em}}$ is the first term of the expansion.

Find and graph $\text{\hspace{0.17em}}{f}_{2}\left(x\right),\text{\hspace{0.17em}}$ such that $\text{\hspace{0.17em}}{f}_{2}\left(x\right)\text{\hspace{0.17em}}$ is the sum of the first two terms of the expansion.

${f}_{2}\left(x\right)={x}^{4}+12{x}^{3}$

Find and graph $\text{\hspace{0.17em}}{f}_{3}\left(x\right),\text{\hspace{0.17em}}$ such that $\text{\hspace{0.17em}}{f}_{3}\left(x\right)\text{\hspace{0.17em}}$ is the sum of the first three terms of the expansion.

Find and graph $\text{\hspace{0.17em}}{f}_{4}\left(x\right),\text{\hspace{0.17em}}$ such that $\text{\hspace{0.17em}}{f}_{4}\left(x\right)\text{\hspace{0.17em}}$ is the sum of the first four terms of the expansion.

${f}_{4}\left(x\right)={x}^{4}+12{x}^{3}+54{x}^{2}+108x$

Find and graph $\text{\hspace{0.17em}}{f}_{5}\left(x\right),\text{\hspace{0.17em}}$ such that $\text{\hspace{0.17em}}{f}_{5}\left(x\right)\text{\hspace{0.17em}}$ is the sum of the first five terms of the expansion.

Extensions

In the expansion of $\text{\hspace{0.17em}}{\left(5x+3y\right)}^{n},\text{\hspace{0.17em}}$ each term has the form successively takes on the value $\text{\hspace{0.17em}}0,1,2,\text{\hspace{0.17em}}...,\text{\hspace{0.17em}}n.$ If $\text{\hspace{0.17em}}\left(\begin{array}{c}n\\ k\end{array}\right)=\left(\begin{array}{c}7\\ 2\end{array}\right),\text{\hspace{0.17em}}$ what is the corresponding term?

$590,625{x}^{5}{y}^{2}$

In the expansion of $\text{\hspace{0.17em}}{\left(a+b\right)}^{n},\text{\hspace{0.17em}}$ the coefficient of $\text{\hspace{0.17em}}{a}^{n-k}{b}^{k}\text{\hspace{0.17em}}$ is the same as the coefficient of which other term?

Consider the expansion of $\text{\hspace{0.17em}}{\left(x+b\right)}^{40}.\text{\hspace{0.17em}}$ What is the exponent of $b$ in the $k\text{th}$ term?

$k-1$

Find $\text{\hspace{0.17em}}\left(\begin{array}{c}n\\ k-1\end{array}\right)+\left(\begin{array}{c}n\\ k\end{array}\right)\text{\hspace{0.17em}}$ and write the answer as a binomial coefficient in the form $\text{\hspace{0.17em}}\left(\begin{array}{c}n\\ k\end{array}\right).\text{\hspace{0.17em}}$ Prove it. Hint: Use the fact that, for any integer $\text{\hspace{0.17em}}p,\text{\hspace{0.17em}}$ such that $\text{\hspace{0.17em}}p\ge 1,\text{\hspace{0.17em}}p!=p\left(p-1\right)!\text{.}$

$\left(\begin{array}{c}n\\ k-1\end{array}\right)+\left(\begin{array}{l}n\\ k\end{array}\right)=\left(\begin{array}{c}n+1\\ k\end{array}\right);\text{\hspace{0.17em}}$ Proof:

$\begin{array}{}\\ \\ \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\begin{array}{c}n\\ k-1\end{array}\right)+\left(\begin{array}{l}n\\ k\end{array}\right)\\ =\frac{n!}{k!\left(n-k\right)!}+\frac{n!}{\left(k-1\right)!\left(n-\left(k-1\right)\right)!}\\ =\frac{n!}{k!\left(n-k\right)!}+\frac{n!}{\left(k-1\right)!\left(n-k+1\right)!}\\ =\frac{\left(n-k+1\right)n!}{\left(n-k+1\right)k!\left(n-k\right)!}+\frac{kn!}{k\left(k-1\right)!\left(n-k+1\right)!}\\ =\frac{\left(n-k+1\right)n!+kn!}{k!\left(n-k+1\right)!}\\ =\frac{\left(n+1\right)n!}{k!\left(\left(n+1\right)-k\right)!}\\ =\frac{\left(n+1\right)!}{k!\left(\left(n+1\right)-k\right)!}\\ =\left(\begin{array}{c}n+1\\ k\end{array}\right)\end{array}$

Which expression cannot be expanded using the Binomial Theorem? Explain.

• $\left({x}^{2}-2x+1\right)$
• ${\left(\sqrt{a}+4\sqrt{a}-5\right)}^{8}$
• ${\left({x}^{3}+2{y}^{2}-z\right)}^{5}$
• ${\left(3{x}^{2}-\sqrt{2{y}^{3}}\right)}^{12}$

The expression $\text{\hspace{0.17em}}{\left({x}^{3}+2{y}^{2}-z\right)}^{5}\text{\hspace{0.17em}}$ cannot be expanded using the Binomial Theorem because it cannot be rewritten as a binomial.

x=-b+_Гb2-(4ac) ______________ 2a
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
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Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
the sum of any two linear polynomial is what
Momo
how can are find the domain and range of a relations
the range is twice of the natural number which is the domain
Morolake