Expand (3x-y)^8

Question

Expand (3x-y)⁸

Answer 1

We can use Pascal's triangle to determine the coefficients of the expansion

of (a+b)ⁿ

 

Exponent "n":                                    Coefficients:

    0                                                         1

    1                                                     1       1

    2                                                  1      2      1

    3                                               1     3      3      1

    4                                            1     4     6      4      1

    5                                        1     5     10    10     5      1

    6                                     1    6     15     20    15     6      1

    7                                  1    7    21     35    35     21     7      1

    8                               1    8    28    56     70    56     28    8      1

 

Note that on each row the numbers are the sum of the two above to the left

and right.  Always with a "1" on the outside.  These numbers represent the

coefficients of the expanded term.

 

(a+b)⁸ = a⁸b⁰ + 8a⁷b + 28a⁶b² + 56a⁵b³+70a⁴b⁴ + 56a³b⁵ + 28a²b⁶ + 8ab⁷ +a⁰b⁸

 

Note you input each term with the power decreasing for the a term and increasing

for the b term up to the power given on the original polynomial being expanded.

The exponents always add up to "n" which in this case is 8

 

In our given polynomial (3x-y)⁸  we have a=3x,  b = -y , n=8

 

(3x-y)⁸ = (3x)⁸(-y)⁰ + 8(3x)⁷(-y)¹ + 28(3x)⁶(-y)² + 56(3x)⁵(-y)³ + 70(3x)⁴(-y)⁴

              + 56(3x)³(-y)⁵ + 28(3x)²(-y)⁶ + 8(3x)¹(-y)⁷ + (3x)⁰(-y)⁸

 

= 6561x⁸ - 17496x⁷y + 20412x⁶y² - 13608x⁵y³ + 5670x⁴y⁴ - 1512x³y⁵

  + 252x²y⁶ - 24xy⁷ + y⁸

 

Because your 2nd term is (-y) the terms in the expansion alternate sign...

(-y)ⁿ = yⁿ when n is even and -yⁿ when n is odd

 

I hope this helps.  Good luck with your class.

Answer 2

Hi Kayla, there are a number of ways to expand (3x - y)⁸, some are more difficult than others. As an example, we could multiply (3x - y) * (3x - y) ... * (3x - y) to get the solution. The problem with this method is keeping the terms in order as the multiplication increases.

 

Another way is to use the Binomial Theorem, which also has several versions.

 

The binomial Theorem using factorial notation is:

 

(a + b)ⁿ = (ⁿ₀) aⁿb⁰ + (ⁿ₁) a^n-1b¹ + (ⁿ₂) a^n-2b² + ... + (ⁿ_n) a⁰bⁿ

 

Notice, the exponents will always add up to n.

 

Here are the first two terms for (3x - y)⁸

 

(3x - y)⁸ = (⁸₀) (3x)⁸(-y)⁰ + (⁸₁) (3x)⁷(-y)¹ + ...

 

The first term expands to: 1 * (3x)⁸ * 1 = (3x)⁸

 

The second term expands to : 8 * (3x)⁷ * (-y)¹ = -8 * (3x)⁷ * y

 

You can follow the expansion of each of the remaining terms, there are seven of them. Check the internet for the Binomial Theorem using factorials.

 

Questions?