Section 1
The Binomial Theorem
Find each value of $x$.$\log _4 16=x$
Find each value of $x$.$\log _x 49=2$
Find each value of $x$.$\log _{25} x=\frac{1}{2}$
Find each value of $x$.$\log _{1 / 2} \frac{1}{8}=x$
Fill in the blanks. Every binomial expansion has _______ more term than the power of the binomial.
Fill in the blanks.The first term in the expansion of $(a+b)^{20}$ is _______
Fill in the blanks.The triangular array that can be used to find the coefficients of a binomial expansion is called _______ triangle.
Fill in the blanks.The symbol 5 ! is read as " _______ "
Fill in the blanks.$6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1=$ _______(Write your answer in factorial notation.)
Fill in the blanks.$8!=8$. _______
Fill in the blanks.$0!=$ _______
Fill in the blanks.According to the binomial theorem, the third term of the expansion of $(a+b)^n$ is _______
Evaluate each expression.3 !
Evaluate each expression.7 !
Evaluate each expression.$-5!$
Evaluate each expression.-6 !
Evaluate each expression.$3!+4$ !
Evaluate each expression.$2!(3!)$
Evaluate each expression.3 !(4!)
Evaluate each expression. $4!+4$ !
Evaluate each expression.$8(7!)$
Evaluate each expression.$4!(5)$
Evaluate each expression. $\frac{9!}{11!}$
Evaluate each expression.$\frac{13!}{10!}$
Evaluate each expression.$\frac{49!}{47!}$
Evaluate each expression.$\frac{101!}{100!}$
Evaluate each expression.$\frac{5!}{3!(5-3)!}$
Evaluate each expression.$\frac{6!}{4!(6-4)!}$
Evaluate each expression. $\frac{7!}{5!(7-5)!}$
Evaluate each expression.$\frac{8!}{6!(8-6)!}$
Evaluate each expression. $\frac{5!(8-5)!}{4!\cdot 7!}$
Evaluate each expression.$\frac{6!\cdot 7!}{(8-3)!(7-4)!}$
Use the binomial theorem to expand each expression.$(x+y)^3$
Use the binomial theorem to expand each expression. $(x+y)^4$
Use the binomial theorem to expand each expression.$(x-y)^4$
Use the binomial theorem to expand each expression.$(x-y)^3$
Use the binomial theorem to expand each expression.$(2 x+y)^3$
Use the binomial theorem to expand each expression.$(x+2 y)^3$
Use the binomial theorem to expand each expression.$(x-2 y)^3$
Use the binomial theorem to expand each expression.$(2 x-y)^3$
Use the binomial theorem to expand each expression.$(2 x+3 y)^3$
Use the binomial theorem to expand each expression.$(3 x-2 y)^3$
Use the binomial theorem to expand each expression.$\left(\frac{x}{2}-\frac{y}{3}\right)^3$
Use the binomial theorem to expand each expression.$\left(\frac{x}{3}+\frac{y}{2}\right)^3$
Use the binomial theorem to expand each expression.$(3+2 y)^4$
Use the binomial theorem to expand each expression. $(2 x+3)^4$
Use the binomial theorem to expand each expression.$\left(\frac{x}{3}-\frac{y}{2}\right)^4$
Use the binomial theorem to expand each expression. $\left(\frac{x}{2}+\frac{y}{3}\right)^4$
Without referring to the text, write the first ten rows of Pascal's triangle.
Find the sum of the numbers in each row of the first ten rows of Pascal's triangle. What is the pattern?
Find the sum of the numbers in the designated diagonal rows of Pascal's triangle shown in the illustration. What is the pattern?
Tell how to construct Pascal's triangle.
Tell how to find the variables of the terms in the expansion of $(r+s)^4$.
If we apply the pattern of the coefficients to the coefficient of the first term in a binomial expansion, the coefficient would be $\frac{n^{\prime}}{0(n-6)}$. Show that this expression is 1 .
If we apply the pattern of the coefficients to the coefficient of the last term in a binomial expansion, the coefficient would be $\frac{n!}{n y(n-n)}$. Show that this expression is 1 .