Squaring a Trinomial

We have learned about numbers and their squares.

For example, if you were asked to find the square of 8, you would immediately say 64 because 8 x 8 = 64. 

However, what answer would you give if someone asked you to square a polynomial? 

You will multiply the polynomial by itself.

Let us take the example of  

\[(x + y)^2\].

This becomes: 

\[(x+y)\times(x+y)\]

\[\therefore x^2 + 2xy + y^2\]

In this mini-lesson, we will explore how we can square a trinomial, which is a type of polynomial.

Lesson Plan

Before learning about trinomials,  let us first understand what a polynomial is.

Polynomial

A polynomial is a mathematical expression written in the form:

\[a_0x^n + a_1x^{n-1} + a_2x^{n-2}...........+ a_nx^0\]

The above expression is also called polynomials in the standard form.

Where \(a_0, a_1, a_2.........a_n\) are constants and \(n\) is a natural number.

Trinomial

A trinomial meaning in math is, it is a type of polynomial that contains only three terms.

The expressions \(x^2 + 2x + 3\),  \(5x^4 - 4x^2 +1\) and \(7y - \sqrt{3} - y^2\) are trinomial examples.

The above trinomial examples are the examples with one variable only, let's take a few more trinomial examples with multiple variables.

\(x^2 + y^2 + xy\),  \(5x^4 - 4x^2 + z\) and \(xyz^3 + x^2z^2 + zy^3\) are trinomials with multiple variables.

Let's recall the names used to refer to polynomials with a different number of terms.

Number of Terms	Example	Polynomial
1	\(xy\)	Monomial
2	\(x + y\)	Binomial
3	\(x^2 + xz + 1\)	Trinomial

Polynomials with more than 3 terms are simply referred to as "Polynomials".

There are no such special names for these types of polynomials.

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Let's take a general trinomial in one variable.

\[P(x) = ax^2 + bx + c\]

Now, if we square a trinomial it means that we are squaring \(P(x)\).

\[{P(x)}^2\]

\[{(ax^2 + bx + c)}^2\]

The above expression means that we are multiplying the trinomial to itself once.

\[{(ax^2 + bx + c)}^2 = (ax^2 + bx + c) \times (ax^2 + bx + c)\]

Now, multiply each term of the first trinomial to the second trinomial.

\({(ax^2 + bx + c)}^2\), on expanding this expression can be written as:

\[{a^2}{x^4} + abx^3 + acx^2 + abx^3 + {b^2}{x^2} + bcx + acx^2 + bcx + c^2\]

Combine the like terms.

\[{a^2}{x^4} + (abx^3 + abx^3) + (acx^2 + acx^2 + {b^2}{x^2}) + (bcx + bcx) + c^2\]

\[{a^2}{x^4} + 2abx^3 + (2acx^2 + {b^2}{x^2}) + 2bcx + c^2\]

\[{a^2}{x^4} + 2abx^3 + (2ac + b^2)x^2 + 2bcx + c^2\]

The above result can be refferd as the squaring a trinomial formula.

Let's take one example of squaring a trinomial.

Example

Find the square of the trinomial \(P(x) = x^2 + 2x + 3\)

Solution

If we compare the above trinomial with a general trinomial in one variable we can say that:

\(a = 1, b = 2, c = 3\)

So, \({(x^2 + 2x + 3)}^2\) can be calculated by putting the values of \(a, b, c\) in the result of squaring a trinomial.

\[{a^2}{x^4} + 2abx^3 + (2ac + b^2)x^2 + 2bcx + c^2\]

\[{1^2}{\times}{x^4} + 2{\times}1{\times}2{\times}x^3 + (2{\times}1{\times}3 + 2^2)x^2 + 2{\times}2{\times}3{\times}x + 3^2\]

\[x^4 + 4x^3 + (6 + 4)x^2 + 12{\times}x + 9\]

\[x^4 + 4x^3 + 10x^2 + 12x + 9\]

We usually do no prefer to remember this big formula, so for finding the square of any trinomial we prefer using the actual multiplication of two trinomials.

Let's take one more example of squaring a trinomial.

Example

Find the square of the trinomial \(P(x) = xy + x + 1\)

Solution

\({(xy + x + 1)}^2\) can be calculated by multiplying the polynomial to itself twice.

\[{(xy + x + 1)}^2 = (xy + x + 1){\times}(xy + x + 1)\]

\[xy(xy + x + 1) + x(xy + x + 1) + 1(xy + x + 1)\]

\[xy{\times}xy + xy{\times}x + xy{\times}1 + x{\times}xy + x{\times}x + x{\times}1 + 1{\times}xy + 1{\times}x + 1{\times}1\]

\[{x^2}{y^2} + {x^2}y + xy + {x^2}y + x^2 + x + xy + x + 1\]

\[{x^2}{y^2} + 2{x^2}y + 2xy + x^2 + 2x + 1\]

\[{x^2}{y^2} + (2y + 1){x^2} + 2xy + 2x + 1\]

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We have learn how to square a trinomial by multiplying a trinomial to itself, let's now explore the squaring a trinomial calculator to understand the squaring of trinimials into more detail.

Enter a trinomial in the simulation below to find out its square.

Example 2

 

 

Paul says that the coefficient of \(xy\) in the square of the trinomial \(x^2y + x + y\) is 1. Is he right?

Solution

To find out whether Paul is right or not let's square the trinomial \(x^2y + x + y\)

The square of the trinomial \(x^2y + x + y\) is \({(x^2y + x + y)}^2\)

\[\begin{align}{({x^2}y + x + y)}^2 &= (x^2y + x + y)(x^2y + x + y) \\[0.2cm]
&= x^2y(x^2y + x + y) + x(x^2y + x + y) + y(x^2y + x + y) \\[0.2cm]
&= (x^4y^2 + x^3y + x^2y^2) + (x^3y + x^2 + xy) + (x^2y^2 + xy + y^2) \\[0.2cm]
&= x^4y^2 + (x^3y + x^3y) + (x^2y^2 + x^2y^2) + x^2 + y^2 + (xy + xy) \\[0.2cm]
&= x^4y^2 + 2x^3y + 2x^2y^2 + x^2 + y^2 + 2xy \\[0.2cm]
&= x^4y^2 + x^2 + y^2 + 2x^3y + 2x^2y^2 + 2xy \end{align}\]

Coefficient of \(xy\) in the square of the trinomial \(x^2y + x + y\) is 2

\(\therefore\) Paul is incorrect.