Section 1
The Binomial Theorem
In the expansion of a binomial, there will be one more term _____________ than the of the binomial.
The _________________ of each term in a binomial expansion is the same as the exponent of the binomial.
The _________ term in a binomial expansion is the first term raised to the power of the binomial.
In the expansion of $(p+q)^{\pi},$ the decrease by 1 in each successive term.decrease by I in cach successive term.Expand 71
Expand 7:____________
Expand 7:_________-$$\begin{aligned}&0 !=\\&1\end{aligned}$$
$$n \cdot \quad=n !$$
In the seventh term of $(a+b)^{11},$ the exponent on $a$ is ___________.
Evaluate each expression.$5 !$$
Evaluate each expression. $$.-5 !$$
Evaluate each expression.$$3 ! \cdot 6 !$$
Evaluate each expression.$$0 ! \cdot 7 !$$
$$6 !+6 !$$
Eraluate each expression.$$\frac{9 !}{12 !}$$
Eraluate each expression.$$\frac{8 !}{5 !}$$
Eraluate each expression.$$\frac{5 ! \cdot 7 !}{9 !}$$
EValuate each expression.$$\frac{3 ! \cdot 5 ! \cdot 7 !}{1 ! 8 !}$$
EValuate each expression.$$\frac{18 !}{6 !(18-6) !}$$
EValuate each expression$$\frac{15 !}{9 !(15-9) !}$$
Use Pascals Triangle to expand each binomial.$$(a+b)^{5}$$
Use Pascals Triangle to expand each binomial.$$(a+b)^{7}$$
Use Pascals Triangle to expand each binomial.$$(x-y)^{3}$$
Use Pascals Triangle to expand each binomial.$$(x-y)^{7}$$
Use the Binomial Theorem to expand each binomial.$$(a+b)^{3}$$
Use the Binomial Theorem to expand each binomial.$$(a+b)^{4}$$
Use the Binomial Theorem to expand each binomial.$$$(a-b)^{5}$$
Use the Binomial Theorem to expand each binomial.$$(x-y)^{4}$$
Use the Binomial Theorem to expand each binomial.$$(2 x+y)^{3}$$
Use the Binomial Theorem to expand each binomial.$$(x+2 y)^{3}$$
Use the Binomial Theorem to expand each binomial.$$(x-2 y)^{3}$$
Use the Binomial Theorem to expand each binomial.$$(2 x-y)^{3}$$
Use the Binomial Theorem to expand each binomial.$$(2 x+3 y)^{4}$$
Use the Binomial Theorem to expand each binomial.$$(2 x-3 y)^{4}$$
Use the Binomial Theorem to expand each binomial.$$(x-2 y)^{4}$$
Use the Binomial Theorem to expand each binomial.$$(x+2 y)^{4}$$
Use the Binomial Theorem to expand each binomial.$$(x-3 y)^{5}$$
Use the Binomial Theorem to expand each binomial.$$(3 x-y)^{5}$$
Use the Binomial Theorem to expand each binomial.$$\left(\frac{x}{2}+y\right)^{4}$$
Use the Binomial Theorem to expand each binomial.$$\left(x+\frac{y}{2}\right)^{4}$$
Find the required term in each binomial expansion.41. $(a+b)^{4} ; 3$ rd term$$(a-b)^{4} ; 2$ nd term$$
Find the required term in each binomial expansion.41. $(a+b)^{4} ; 3$ rd term$$(a+b)^{7} ; 5$ th term$$
Find the required term in each binomial expansion.41. $(a+b)^{4} ; 3$ rd term$$(a+b)^{5} ; 4$ th term$$
Find the required term in each binomial expansion.41. $(a+b)^{4} ; 3$ rd term$$(a-b)^{5} ;$ 6th term$$
Find the required term in each binomial expansion.41. $(a+b)^{4} ; 3$ rd term$$(a-b)^{8} ; 7$ th term$$
Find the required term in each binomial expansion.41. $(a+b)^{4} ; 3$ rd term$$(a+b)^{17} ; 5$ th term$$
Find the required term in each binomial expansion.41. $(a+b)^{4} ; 3$ rd term$$(a-b)^{12} ; 3$ rd term$4
Find the required term in each binomial expansion.41. $(a+b)^{4} ; 3$ rd term$$(a-\sqrt{2})^{4} ; 2$ nd term$$
Find the required term in each binomial expansion.41. $(a+b)^{4} ; 3$ rd term$$(a-\sqrt{3})^{s} ; 3 r d$ term$4
Find the required term in each binomial expansion.41. $(a+b)^{4} ; 3$ rd term$(a+\sqrt{3} b)^{9} ; 5$ th term
Find the required term in each binomial expansion.41. $(a+b)^{4} ; 3$ rd term$(\sqrt{2} a-b)^{7} ; 4$ th term
Find the required term in each binomial expansion.41(a-\sqrt{3})^{8} ; 3 \mathrm{rd} \text { term }
Find the required term in each binomial expansion.41$$\left(\frac{x}{2}+y\right)^{4} ; 3$ rd term
Find the required term in each binomial expansion.41$$\left(m+\frac{n}{2}\right)^{8} ; 3$ rd term
Find the required term in each binomial expansion.41$$\left(\frac{r}{2}-\frac{s}{2}\right)^{11} ; 10$ th term
Find the required term in each binomial expansion.41$$\left(\frac{p}{2}-\frac{q}{2}\right)^{9} ;$ oth term
Find the required term in each binomial expansion.41$$(a+b)^{n .} ; 4$ th term
Find the required term in each binomial expansion.41$$(a-b)^{n} ; 5$ th term
Find the required term in each binomial expansion.41$$(a+b)^{n-} ; r$ th term
Find the required term in each binomial expansion.41$$(a+b)^{n} ;(r+1)$ th term
61. Find the sum of the numbers in each row of the first ten rows of Pascal's Triangle. Do you sce a pattern?
Show that the sum of the coefficients in the binomial expansion of $(x+y)^{n}$ is $2^{n}$. (Hint: Let $x=\bar{y}=1 .)$
If we applied the pattern of coefficients to the coefficient of the first term in the Binomial Theorem, it would be $\frac{n !}{0 !(n-0) !}$. Show that this expression equals 1.
If we applicd the pattern of coefficients to the coef ficient of the last term in the Binomial Theorem, it would be $\frac{n !}{n !(n-n) !}$. Show that this expression
Define factorial notation and explain how to evaluate $10 !$
With a calculator, evaluate $69 !$. Explain why we cannot find $70 !$ with a calculator.
Explain how the $r$ th term of a binomial expansion is constructed.
Explain the four patterns apparent in a binomial expansion.
Facter each expression.$$3 x^{3} y^{2} z^{4}-6 x y z^{5}+15 x^{2} y z^{2}$$
Facter each expression.$$3 z^{2}-15 t z+12 t^{2}$$
Facter each expression.$$a^{4}-b^{4}$$
Facter each expression.$$3 r^{4}-36 r^{3}-135 r^{2}$$
Simplify each complex fraction.frac{\frac{1}{x}+\frac{1}{3}}{\frac{1}{x}-\frac{1}{3}}$$
$$\text { 76. } \frac{x-\frac{1}{y}}{y-\frac{1}{x}}$$