Summary
The binomial theorem is a shortcut that we can use to expand expressions that have binomial bases raised to an exponent.
If you were to simplify a binomial power like\((2x-9)^7\) the long way, you would have to use polynomial multiplication to multiply \((2x-9)\) by itself 7 times. That would take a LONG time.
The binomial expansion process is much faster and it is pretty easy once you get the hang of it.
\[(x+y)^n= \sum_{k=0}^{n} {n \choose k}{x^{n-k}}{y^{k}}\]
The standard formula for the binomial theorem uses summation notation and combination notation.
If you haven’t learned these notations yet, you can still use the binomial theorem by following the patterns shown in these binomial expansions:
\[(x+y)^2=1x^2+2xy+1y^2\]
\[(x+y)^3=1x^3+3x^2y+3xy^2+1y^3\]
\[(x+y)^4=1x^4+4x^3y+6x^2y^2+4xy^3+1y^4\]
\[(x+y)^5=1x^5+5x^4y+10x^3y^2+10x^2y^3+5xy^4+1y^5\]
What patterns do you notice?
There are 4 important patterns that show how you can use the exponent of any binomial power to expand it into a polynomial.
When the binomial power has an exponent of n…
- There are n+1 terms in the expanded polynomial.
- The coefficients match the nth row of Pascal’s Triangle
- The powers of x decrease from n to 0.
- The powers of y increase from 0 to n.
As an example, look at the expansion of \((x+y)^4\).
\[(x+y)^4={\blue1}{\yellow x^4}{\green y^0}+{\blue4}{\yellow x^3}{\green y^1}+{\blue6}{\yellow x^2}{\green y^2}+{\blue4}{\yellow x^1}{\green y^3}+{\blue1}{\yellow x^0}{\green y^4}\]
The exponent of the binomial power is a 4. If we add 1 to that exponent, it tells us that there are 5 terms in the expanded polynomial.
The coefficients of those terms match the 4th row of Pascal’s Triangle. And the powers of x decrease from 4 to 0 while the powers of y increase from 0 to 4.
How to Use the Binomial Theorem
The binomial theorem will expand any power as long as the base of the exponent is a binomial (a polynomial with exactly 2 terms).
- Determine the number of terms by adding 1 to the exponent.
- Use Pascal’s Triangle to find the coefficients of each term.
- List the first term of the binomial with decreasing powers.
- List the second term of the binomial with increasing powers.
- Simplify each term by simplifying the exponents and multiplying the coefficients if there are multiple coefficients in each term.
In the expansions above, I expanded the binomial powers of \((x+y)\). But you can also use the binomial theorem to expand powers of more complicated binomials like \((3x+8y)\) or \((x^2-8)\) or \((7x^5-4x^2)\).
If the terms of the binomial have multiple parts (coefficients, variables, exponents, etc.), then you need to put them inside of parentheses when you list them with decreasing/increasing powers.
You also have to follow the power rule and the distributive property for exponents when you simplify the terms of the expanded polynomial.
Example
Expand: \[(3x+2)^4\]
How many terms are there in the binomial expansion?
\[({\yellow3x}+{\green 2})^{\blue 4}\]
The exponent in this binomial power is a 4. The number of terms is always 1 more than the exponent. So, that means that I will have 5 terms in my expanded polynomial.
What are the coefficients of the terms?
\[({\yellow3x}+{\green 2})^{\blue 4}\]
Because the exponent of the binomial power is a 4, I’ll use the 4th row of Pascal’s Triangle to find the coefficients of the expanded polynomial.
\[{\blue1}\_\_+{\blue4}\_\_+{\blue6}\_\_+{\blue4}\_\_+{\blue1}\_\_\]
What are the expanded powers of the first term of the binomial?
\[({\yellow3x}+{\green 2})^{\blue 4}\]
The monomial 3x is the first term of the binomial. To expand the first term, I need to write powers of 3x with decreasing exponents from 4 to 0 in each term.
\[{\blue1}{\yellow (3x)^4}\_\_+{\blue4}{\yellow (3x)^3}\_\_+{\blue6}{\yellow (3x)^2}\_\_+{\blue4}{\yellow (3x)^1}\_\_+{\blue1}{\yellow (3x)^0}\_\_\]
What are the expanded powers of the second term of the binomial?
\[({\yellow3x}+{\green 2})^{\blue 4}\]
The constant 2 is the second term of the binomial. To expand the second term, I need to write powers of 2 with increasing exponents from 0 to 4 in each term.
\[{\blue1}{\yellow (3x)^4}{\green 2^0}+{\blue4}{\yellow (3x)^3}{\green 2^1}+{\blue6}{\yellow (3x)^2}{\green 2^2}+{\blue4}{\yellow (3x)^1}{\green 2^3}+{\blue1}{\yellow (3x)^0}{\green 2^4}\]
What is the most simplified form of each term?
To simplify each of the terms in the polynomial, I’ll simplify the exponents and multiply the coefficients.
\[{\blue1}{\yellow (3x)^4}{\green 2^0}={\blue1}\cdot{\yellow 81x^4}\cdot{\green 1}=81x^4\]
\[{\blue4}{\yellow (3x)^3}{\green 2^1}={\blue4}\cdot{\yellow 27x^3}\cdot{\green 2}=216x^3\]
\[{\blue6}{\yellow (3x)^2}{\green 2^2}={\blue6}\cdot{\yellow 9x^2}\cdot{\green 4}=216x^2\]
\[{\blue4}{\yellow (3x)^1}{\green 2^3}={\blue4}\cdot{\yellow 3x}\cdot{\green 8}=96x\]
\[{\blue1}{\yellow (3x)^0}{\green 2^4}={\blue1}\cdot{\yellow 1}\cdot{\green 16}=16\]
The expanded form of \((3x+2)^4\) is…\[81x^4+216x^3y+216x^2y^2+96x+16\]
