How To Square A Trinomial

Nosotros accept learned about numbers and their squares.

For case, if y'all were asked to find the square of 8, you would immediately say 64 because viii x 8 = 64.

Nonetheless, what respond would you give if someone asked you to square a polynomial?

You will multiply the polynomial by itself.

Let us take the case of

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

This becomes:

\[(ten+y)\times(10+y)\]

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

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

Lesson Programme

What Is a Trinomial?

Before learning most trinomials, permit us outset sympathize what a polynomial is.

Polynomial

A polynomial is a mathematical expression written in the form:

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

The above expression is also chosen polynomials in the standard grade.

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 simply iii terms.

terms of polynomials

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

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

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

Allow'south 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	\(ten^2 + xz + 1\)	Trinomial

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

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

How to Foursquare a Trinomial?

Let's take a general trinomial in one variable.

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

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 ways that we are multiplying the trinomial to itself once.

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

At present, multiply each term of the first trinomial to the 2nd trinomial.

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

expanding and squaring a trinomial

terms of polynomials

\[{a^2}{10^4} + abx^3 + acx^two + abx^three + {b^2}{ten^2} + bcx + acx^2 + bcx + c^2\]

Combine the like terms.

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

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

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

The higher up effect can be refferd equally the squaring a trinomial formula.

Allow'due south accept one example of squaring a trinomial.

Example

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

Solution

If we compare the above trinomial with a full general trinomial in i variable we tin say that:

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

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

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

\[{1^2}{\times}{ten^4} + ii{\times}1{\times}2{\times}ten^iii + (ii{\times}one{\times}3 + 2^2)ten^2 + two{\times}two{\times}three{\times}x + 3^two\]

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

\[x^4 + 4x^3 + 10x^2 + 12x + nine\]

We usually practice no prefer to call up this big formula, and so for finding the square of any trinomial we adopt using the bodily multiplication of two trinomials.

Permit's take 1 more example of squaring a trinomial.

Example

Detect the foursquare of the trinomial \(P(x) = xy + x + 1\)

Solution

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

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

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

\[xy{\times}xy + xy{\times}10 + xy{\times}1 + x{\times}xy + 10{\times}x + x{\times}1 + ane{\times}xy + i{\times}10 + ane{\times}ane\]

\[{10^2}{y^2} + {x^two}y + xy + {10^2}y + x^2 + ten + xy + x + i\]

\[{x^2}{y^two} + 2{x^2}y + 2xy + x^2 + 2x + ane\]

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

Squaring a Trinomial Calculator

We accept acquire how to foursquare a trinomial by multiplying a trinomial to itself, permit's now explore the squaring a trinomial estimator to understand the squaring of trinimials into more particular.

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

important notes to remember

Important Notes

The degree of a polynomial is the highest ability of the variable.
If the degree of trinomial is \(n\) then the degree of its square is \(n^2\)
A trinomial with 1 variable and degree 2 is also known as a quadratic trinomial.

Solved Examples on Squaring a Trinomial

Assist Tom to observe the square of the trinomial \(10 + y + z\).

Solution

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

\[\begin{align}{(x + y + z)}^2 &= (x + y + z)(x + y + z) \\[0.2cm]
&= x(x + y + z) + y(10 + y + z) + z(x + y + z) \\[0.2cm]
&= (ten^2 + xy + xz) + (xy + y^2 + yz) + (xz + yz + z^2) \\[0.2cm]
&= x^2 + y^ii + z^2 + (xy + xy) + (yz + yz) + (xz + xz) \\[0.2cm]
&= x^2 + y^2 + z^2 + 2xy + 2yz + 2xz \\[0.2cm]
&= x^2 + y^2 + z^ii + ii(xy + yz + xz) \end{marshal}\]

\(\therefore\) \(x^2 + y^two + z^2 + ii(xy + yz + xz)\)

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

Solution

To observe out whether Paul is right or non permit'due south square the trinomial \(ten^2y + x + y\)

The square of the trinomial \(x^2y + x + y\) is \({(10^2y + 10 + y)}^two\)

\[\begin{align}{({x^2}y + x + y)}^two &= (x^2y + 10 + y)(x^2y + ten + y) \\[0.2cm]
&= x^2y(10^2y + x + y) + ten(x^2y + x + y) + y(x^2y + x + y) \\[0.2cm]
&= (x^4y^ii + x^3y + x^2y^ii) + (x^3y + ten^2 + xy) + (x^2y^2 + xy + y^2) \\[0.2cm]
&= x^4y^2 + (ten^3y + x^3y) + (10^2y^2 + ten^2y^2) + 10^2 + y^two + (xy + xy) \\[0.2cm]
&= 10^4y^2 + 2x^3y + 2x^2y^2 + x^ii + y^two + 2xy \\[0.2cm]
&= x^4y^2 + 10^two + y^two + 2x^3y + 2x^2y^2 + 2xy \end{align}\]

Coefficient of \(xy\) in the square of the trinomial \(x^2y + 10 + y\) is 2

\(\therefore\) Paul is incorrect.

Challenge your math skills

Challenging Questions

Find the coefficent of \(x^iv\) in the expansion of \({(x + i)}^6\).
Find the square root of the trinomial \(81x^2 + 126xy + 49y^two\).

Interactive Questions on Squaring a Trinomial

Here are a few activities for yous to practice.

Select/Type your answer and click the "Check Answer" push to run into the result.

Let'due south Summarize

The mini-lesson targeted the fascinating concept of squaring a trinomial. The math journey effectually squaring a trinomial starts with what a pupil already knows, and goes on to creatively crafting a fresh concept in the immature minds. Done in a way that is non only relatable and easy to grasp, but will besides stay with them forever. Here lies the magic with Cuemath.

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Be it worksheets, online classes, uncertainty sessions, or whatever other form of relation, information technology's the logical thinking and smart learning approach that we, at Cuemath, believe in.

Frequently Asked Questions (FAQs)

1. What is a trinomial in math?

A trinomial meaning in math is, information technology is a type of polynomial that contains only three terms.
For instance, \(x - 2y - z\) or \( x^2 + y^2 + xy\)

ii. What is a trinomial square?

Let's take a trinomial \(P(x) = x + y + xy\).
So, a trinomial square is the square of \(P(10)\).
\({P(10)}^two\) is the trinomial square for \(P(ten)\).
\({P(x)}^two = {(x + y + xy)}^2 = (x + y + xy)(x + y + xy)\)

3.Which polynomial is a perfect square trinomial?

Polynomials like \(x^2 + 2x + 1\) is an example of perfect foursquare trinomial.
\(ten^two + 2x + 1\) can be written as \({(x + i)}^2\).

4. How practice y'all foursquare a trinomial?

A trinomial can be squared past multiplying itself twice and performed the required calculations.
Explore the calculator above to detect out the squaring of a trinomial in more detail.