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Question: How do you expand \( {(r + 3)^5} \) using Pascal’s Triangle?...

How do you expand (r+3)5{(r + 3)^5} using Pascal’s Triangle?

Explanation

Solution

Hint : We will start off by explaining Pascal's Triangle. Then mention its different forms and rules. Then we will mention the expansion. After that we will compare the coefficients and then substitute in the formula and then simplify the terms.

Complete step-by-step answer :
We will start off by displaying Pascal's Triangle.
1 1−1 1−2−1 1−3−3−1 1−4−6−4−1 1−5−10−10−5−1  1 \\\ 1 - 1 \\\ 1 - 2 - 1 \\\ 1 - 3 - 3 - 1 \\\ 1 - 4 - 6 - 4 - 1 \\\ 1 - 5 - 10 - 10 - 5 - 1 \\\
Now we can use this triangle to evaluate the coefficients of the expansion of the term (a+b)n{(a + b)^n} by taking the exponent nn and then adding 11 . Here, the coefficients will correspond with the line n+1n + 1 of the triangle.
Now if we compare the coefficients (r+3)5{(r + 3)^5} we know that here n=5n = 5 so the coefficients of the expansion will correspond with the line 66 .
Here, the expansion follows the rule
(a+b)n=c0anb0+c1an−1b1+cn−1a1bn−1+cna0bn{(a + b)^n} = {c_0}{a^n}{b^0} + {c_1}{a^{n - 1}}{b^1} + {c_{n - 1}}{a^1}{b^{n - 1}} + {c_n}{a^0}{b^n}
Now, the values of the coefficients, from the triangle are,
1−5−10−10−5−11 - 5 - 10 - 10 - 5 - 1
=1a5b0+5a4b+10a3b2+10a2b3+5ab4+1a0b5= 1{a^5}{b^0} + 5{a^4}b + 10{a^3}{b^2} + 10{a^2}{b^3} + 5a{b^4} + 1{a^0}{b^5}
Now we will substitute the actual values of a,ra,r and bb into the expression.
=1(r)5(3)0+5(r)4(3)1+10(r)3(3)2+10(r)2(3)3+5(r)(3)4= 1{(r)^5}{(3)^0} + 5{(r)^4}{(3)^1} + 10{(r)^3}{(3)^2} + 10{(r)^2}{(3)^3} + 5(r){(3)^4}
Now, we will simplify the terms to evaluate the value.
=r5+15r4+90r3+270r2+405r+243= {r^5} + 15{r^4} + 90{r^3} + 270{r^2} + 405r + 243
Hence, the expansion of the expression (r+3)5{(r + 3)^5} is r5+15r4+90r3+270r2+405r+243{r^5} + 15{r^4} + 90{r^3} + 270{r^2} + 405r + 243 .

Note : Pascal’s Triangle is a triangular array constructed by summing adjacent elements in preceding rows. It is a triangular array of the binomial coefficients that arises in probability theory, combinatorics and algebra. The rows of Pascal’s Triangle are conventionally enumerated starting with row n=0n = 0 at the top. The entries in each row are numbered from the left beginning with k=0k = 0 and are usually staggered relative to the numbers in the adjacent rows.
While converting orders do not matter for addition and multiplication. But order is important for subtraction and division. While comparing and substituting the values make sure to compare along with their signs.