Your task as Agent Binomial Expansion, should you choose to accept it, is to expand the binomial \begin{align*}(x-2)^5\end{align*}

### Pascals Triangle

**Pascal’s Triangle:**

Each row begins and ends with a one. Each “interior” value in each row is the sum of the two numbers above it. For example, \begin{align*}2+1=3\end{align*}

Another pattern that can be observed is that the row number is equal to the number of elements in that row. Row 1, for example has 1 element, 1. Row 2 has 2 elements, 1 and 1. Row 3 has 3 elements, 1, 2 and 1.

A third pattern is that the second element in the row is equal to one less than the row number. For example, in row 5 we have 1, 4, 6, 4 and 1.

#### Solve the following problems

Continue the triangle to determine the elements in the \begin{align*}9^{th}\end{align*}

Following the pattern of adding adjacent elements to get the elements in the next row, we find hat the eighth row is: \begin{align*}1 \ \ 7 \ \ 21 \ \ 35 \ \ 35 \ \ 21 \ \ 7 \ \ 1\end{align*}

Now, continue the pattern again to find the \begin{align*}9^{th}\end{align*}

Expand the binomial \begin{align*}(a+b)^4\end{align*}

\begin{align*}&(a+b)(a+b)(a+b)(a+b) \\
&(a^2+2ab+b^2)(a^2+2ab+b^2) \\
a^4+2a^3b+a^2& b^2+2a^3b+4a^2b^2+2ab^3+a^2b^2+2ab^3+b^4 \\
& a^4+4a^3b+6a^2b^2+4ab^3+b^4 \end{align*}

1. Take two binomials at a time and square them using \begin{align*}(a+b)^2=a^2+2ab+b^2\end{align*}

2. Next, distribute each term in the first trinomial over each term in the second trinomial and collect like terms.

We can see that the powers of \begin{align*}a\end{align*}

Use what you discovered in the previous example to expand \begin{align*}(x+y)^6\end{align*}

The degree of this expansion is 6, so the powers of \begin{align*}x\end{align*}

\begin{align*}\underline{\; \; \; \; \;}x^6 \ +\underline{\; \; \; \; \;}x^5 y \ +\underline{\; \; \; \; \;}x^4 y^2 \ +\underline{\; \; \; \; \;}x^3 y^3 \ +\underline{\; \; \; \; \;}x^2 y^4 \ +\underline{\; \; \; \; \;}xy^5 \ +\underline{\; \; \; \; \;}y^6\end{align*}

In the previous example we observed that the coefficients for a fourth degree binomial were found in the fifth row of Pascal’s Triangle. Here we have a \begin{align*}6^{th}\end{align*}

\begin{align*}x^6+6x^5y+15x^4y^2+20x^3y^3+15x^2y^4+6xy^5+y^6\end{align*}

### Examples

#### Example 1

Earlier, you were asked to expand the binomial \begin{align*}(x-2)^5\end{align*}

To expand the binomial \begin{align*}(x-2)^5\end{align*}

The degree of this expansion is 5, so the powers of \begin{align*}x\end{align*}

\begin{align*}\underline{\; \; \; \; \;}x^5 \ +\underline{\; \; \; \; \;}x^4 y \ +\underline{\; \; \; \; \;}x^3 y^2 \ +\underline{\; \; \; \; \;}x^2 y^3 \ +\underline{\; \; \; \; \;}xy^4 \ +\underline{\; \; \; \; \;}y^5\end{align*}

The coefficients for a fifth degree binomial can be found in the sixth row of Pascal’s Triangle. Now we can fill in the blanks with the correct coefficients, replacing *y* with \begin{align*}-2\end{align*}

\begin{align*}1\cdot x^5 + 5x^4(-2) + 10x^3(-2)^2 + 10x^2(-2)^3 + 5x(-2)^4 + 1\cdot (-2)^5\\
x^5-10x^4+40x^3-80x^2+80x-32\end{align*}

#### Example 2

Write out the elements in row 10 of Pascal’s Triangle.

The \begin{align*}9^{th}\end{align*}

Subsequently, the \begin{align*}10^{th}\end{align*}

#### Example 3

Expand \begin{align*}(a+4)^3\end{align*}

\begin{align*}a^3&+3a^2(4)+3a(4)^2+(4)^3 \\
& a^3+12a^2+48a+64\end{align*}

#### Example 4

Write out the coefficients in the expansion of \begin{align*}(2x-3)^4\end{align*}

\begin{align*}(2x)^4+4&(2x)^3(-3)+6(2x)^2(-3)^2+4(2x)(-3)^3+(-3)^4 \\
&16x^4-96x^3+216x^2-216x+81\end{align*}

### Review

- Write out the elements in row 7 of Pascal’s Triangle.
- Write out the elements in row 13 of Pascal’s Triangle.

Use Pascal’s Triangle to expand the following binomials.

- \begin{align*}(x-6)^4\end{align*}
(x−6)4 - \begin{align*}(2x+5)^6\end{align*}
(2x+5)6 - \begin{align*}(3-x)^7\end{align*}
(3−x)7 - \begin{align*}(x^2-2)^3\end{align*}
- \begin{align*}(x+4)^5\end{align*}
- \begin{align*}(2-x^3)^4\end{align*}
- \begin{align*}(a-b)^6\end{align*}
- \begin{align*}(x+1)^{10}\end{align*}

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 12.6.