Question

Find the expanded form of \[{\left[ {p + 2q + r} \right]^2}\].

Answer 1

Hint: We will use the formula for the square of a trinomial to find the expansion of \[{\left[ {p + 2q + r} \right]^2}\]. We will find squares of the three terms and the product of terms taken 2 at a time and substitute the values in the formula.

Complete step-by-step answer:
A trinomial is a polynomial with three terms.
The square of a trinomial is given by the following formula:
\[{\left[ {a + b + c} \right]^2} = {a^2} + {b^2} + {c^2} + 2\left[ {ab + bc + ca} \right]\]
The first term of the given trinomial is \[p\], the second term is \[2q\] and the third term is \[r\].
First, let us find the square of the first term.
\[{\left[ p \right]^2} = {p^2}\]
The square of the first term is \[{p^2}\].
Next, let us find the square of the second term.
\[\begin{array}{c}{\left[ {2q} \right]^2} = {2^2} \cdot {q^2}\\ = 4{q^2}\end{array}\]
The square of the second term is \[4{q^2}\].
Similarly, let us find the square of the third term.
\[{\left[ r \right]^2} = {r^2}\]
The square of the third term is \[{r^2}\].
Now, let us find the product of the terms taken two at a time.
The product of the first 2 terms is \[p \cdot 2q = 2pq\].
The product of the 2nd and the 3rd term is \[2q \cdot r = 2qr\].
The product of the 3rd and the 1st term is \[r \cdot p = pr\].
We will substitute the values that we have calculated in the formula for the square of a trinomial.
Let us substitute \[{p^2}\] for \[{a^2}\], \[4{q^2}\] for \[{b^2}\], \[{r^2}\] for \[{c^2}\], \[2pq\] for \[ab\], \[2qr\] for \[bc\] and \[pr\] for \[ca\] in the formula for the square of a trinomial.
\[{\left[ {p + 2q + r} \right]^2} = {p^2} + 4{q^2} + {r^2} + 2\left[ {2pq + 2qr + pr} \right]\] .
Now, let us solve the bracket in the L.H.S of the above equation.
\[\begin{array}{c}{\left[ {p + 2q + r} \right]^2} = {p^2} + 4{q^2} + {r^2} + 2\left[ {2pq + 2qr + pr} \right]\\ = {p^2} + 4{q^2} + {r^2} + 4pq + 4qr + 2pr\end{array}\]

Hence, the expansion of the given trinomial is \[{p^2} + 4{q^2} + {r^2} + 4pq + 4qr + 2pr\].

Note: We can also find the expansion by multiplying the trinomial with itself.
\[{\left[ {p + 2q + r} \right]^2} = \left[ {p + 2q + r} \right]\left[ {p + 2q + r} \right]\]
According to the distributive property, 2 polynomials can be multiplied in the following way. \[\left[ {a + b + c} \right]\left[ {p + q + r} \right] = a\left[ {p + q + r} \right] + b\left[ {p + q + r} \right] + c\left[ {p + q + r} \right]\]
We will use the distributive property to solve the expression further.
\[\left[ {p + 2q + r} \right]\left[ {p + 2q + r} \right] = p\left[ {p + 2q + r} \right] + 2q\left[ {p + 2q + r} \right] + r\left[ {p + 2q + r} \right]\]
We will again apply the distributive property to each of the three terms in the L.H.S of the equation and simplify the expression.
\[\begin{array}{c}\left[ {p + 2q + r} \right]\left[ {p + 2q + r} \right] = p\left[ {p + 2q + r} \right] + 2q\left[ {p + 2q + r} \right] + r\left[ {p + 2q + r} \right]\\ = p \cdot p + p \cdot 2q + p \cdot r + 2q \cdot p + 2q \cdot 2q + 2q \cdot r + r \cdot p + r \cdot 2q + r \cdot r\\ = {p^2} + 2pq + pr + 2pq + 4{q^2} + 2qr + pr + 2qr + {r^2}\end{array}\]
Now, we will collect the like terms and add them.
\[\begin{array}{c}{\left[ {p + 2q + r} \right]^2} = {p^2} + 2pq + pr + 2pq + 4{q^2} + 2qr + pr + 2qr + {r^2}\\ = {p^2} + \left[ {2pq + 2pq} \right] + \left[ {pr + pr} \right] + 4{q^2} + \left[ {2qr + 2qr} \right] + {r^2}\\ = {p^2} + 4pq + 2pr + 4{q^2} + 4qr + {r^2}\end{array}\]
We have obtained the same expression for the expansion of the given term by this method also.