b Expand the expression by using Pascal's Triangle and the Binomial Theorem.
A
a x^5+15x^4+90x^3+270x^2+420x+243
B
b 16p^4-32p^3q+24p^2q^2-8pq^3+q^4
Practice makes perfect
a To expand the given binomial, we should recall the Binomial Theorem. It states that for every positive integer n, we can expand the expression (a+b)^n by using the numbers in the n^(th) row of Pascal's Triangle.
cc
(a+b)^n= & _n C_0a^nb^0+ _n C_1a^(n-1)b^1
& +
& ...
& +
& _n C_(n-1)a^1b^(n-1)+ _n C_na^0b^n
In the above formula, _n C_0, _n C_1, ..., _n C_n are the numbers in the n^(th) row of Pascal's Triangle.
Row 1.3cm Pascal's Triangle 1.2cm
cccccccccccc
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
Note that each number greater than 1 found in the triangle is the sum of the two numbers diagonally above it. Now consider the given binomial.
( x+ 3 )^5
We can substitute the first term for a and the second term for b using the Binomial Theorem equation and use the coefficients from the {\color{#FF0000}{5}}^\text{th} row of Pascal's Triangle.
b To expand the given binomial, we should recall the Binomial Theorem. It states that for every positive integer n, we can expand the expression (a+b)^n by using the numbers in the n^(th) row of Pascal's Triangle.
cc
(a+b)^n= & _nC_0a^nb^0+ _nC_1a^(n-1)b^1
& +
& ...
& +
& _nC_(n-1)a^1b^(n-1)+ _nC_na^0b^n
In the above formula, _nC_0, _nC_1, ..., _nC_n are the numbers in the n^(th) row of Pascal's Triangle.
Row 1.3cm Pascal's Triangle 1.2cm
cccccccccccc
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
Note that each number greater than 1 found in the triangle is the sum of the two numbers diagonally above it. Now consider the given binomial.
(2p-q)^4 ⇔ ( 2p+( - q) )^4
We can substitute the first term for a and the second term for b using the Binomial Theorem equation and use the coefficients from the {\color{#FF0000}{4}}^\text{th} row of Pascal's Triangle.
