# The Binomial Theorem

Binomial theorem

Tip: This isn't the place to ask a question because the teacher can't reply.

## Key Questions

• Assuming that we need to find a specific term of ${\left(a + b\right)}^{n}$ then
We can use a formula,

${t}_{r + 1} = n {C}_{r} \cdot {a}^{n - r} \cdot {b}^{r}$

For example to find ${t}_{4}$ (fourth term) of ${\left(a + b\right)}^{n}$

${t}_{3 + 1} = n {C}_{3} \cdot {a}^{n - 3} \cdot {b}^{3}$

1. find the ${9}^{t h}$ term of ${\left(x + 5\right)}^{13}$.

As said we can use the formula here too.

${t}_{8 + 1} = 13 {C}_{8} \cdot {x}^{13 - 8} \cdot {5}^{8} = 502734375 {x}^{5}$

See explanation...

#### Explanation:

Binomial theorem is a kind of formula that helps us to expand binomials raised to the power of any number using the pascals triangle or using the binomial theorem.

Watch the video to now about the pascal's triangle and the binomial theorem.

Now we can use the pascal's triangle to solve the following expressions:

${\left(a + b\right)}^{2} = 1 {a}^{2} + 2 a b + 1 {b}^{2} = {a}^{2} + 2 a b + {b}^{2}$

$E g . \left(2 + b\right) = 1 \left({2}^{2}\right) + 2 \left(2 b\right) + 1 \left({b}^{2}\right) = 1 \left(4\right) + 2 \left(2 b\right) + 1 \left({b}^{2}\right) = 4 + 4 b + {b}^{2}$

${\left(a + b\right)}^{3} = 1 {a}^{3} + 3 {a}^{2} b + 3 a {b}^{2} + 1 {b}^{3}$

$E g . {\left(b + 5\right)}^{3} = 1 {b}^{3} + 3 {b}^{2}$$5 + 3 b {5}^{2} + 1 \left({5}^{3}\right) = {b}^{3} + 15 {b}^{2} + 75 b + 125$

• Let ${\left(2 x + 3\right)}^{3}$ be a given binomial.

From the binomial expression, write down the general term. Let this term be the r+1 th term. Now simplify this general term. If this general term is a constant term, then it should not contain the variable x.
Let us write the general term of the above binomial.
${T}_{r + 1}$ = "" ^3 C_r ${\left(2 x\right)}^{3 - r}$ ${3}^{r}$

simplifying, we get, ${T}_{r + 1}$= "" ^3 C_r ${2}^{3 - r}$ ${3}^{r}$ ${x}^{3 - r}$

Now for this term to be the constant term, ${x}^{3 - r}$ should be equal to 1.
Therefore, ${x}^{3 - r}$= ${x}^{0}$
=> 3-r =0
=> r=3

Thus, the fourth term in the expansion is the constant term. By putting r=3 in the general term, we will get the value of the constant term.

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