Sure, let's solve the problem step-by-step.
We are given the expression:
[tex]\[ p^3 - 9p^2q + 27pq^2 - 27q^3 \][/tex]
We can compare this with the identity for the cube of a binomial:
[tex]\[ (a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3 \][/tex]
Let's try to match the terms in the given expression with those in the identity.
First, notice the coefficients in the given expression and how they correspond to the identity:
- The given first term is [tex]\( p^3 \)[/tex], which matches the [tex]\( a^3 \)[/tex] term in the identity.
- The second term is [tex]\( -9p^2q \)[/tex], and in the identity, the coefficient of the [tex]\( a^2b \)[/tex] term is [tex]\(-3\)[/tex]. Therefore, we need to find [tex]\( a \)[/tex] and [tex]\( b \)[/tex] such that [tex]\(-3a^2b = -9p^2q\)[/tex]. From this, we can see that [tex]\( a = p \)[/tex] and [tex]\( b = 3q \)[/tex].
- The third term in the given expression is [tex]\( 27pq^2 \)[/tex], and in the identity, the coefficient of the [tex]\( ab^2 \)[/tex] term is [tex]\( 3 \)[/tex]. We need to verify if the chosen [tex]\( a \)[/tex] and [tex]\( b \)[/tex] satisfy this term too: [tex]\( 3ab^2 = 3(p)(3q)^2 = 3(p)(9q^2) = 27pq^2 \)[/tex], which matches.
- Lastly, the fourth term in the given expression is [tex]\( -27q^3 \)[/tex], and in the identity, the coefficient of the [tex]\( b^3 \)[/tex] term is [tex]\(-1\)[/tex]. We need to verify if the chosen values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] satisfy this term as well: [tex]\( -b^3 = -(3q)^3 = -27q^3\)[/tex], which also matches.
Since all the terms match, we can rewrite the given expression using the binomial cube identity as follows:
[tex]\[ p^3 - 9p^2q + 27pq^2 - 27q^3 = (p - 3q)^3 \][/tex]
Thus, the simplified form of the given expression is:
[tex]\[ (p - 3q)^3 \][/tex]
