Trinomial

Trinomial

In this mini-lesson, we will explore the world of trinomials by understanding different types of trinomials, rules for trinomials, and how to apply them while solving problems. 

We will also discover interesting facts around them.

We have been learning about polynomials from a very young age.

When we were first introduced to variables and expressions, we were simply dealing with polynomials. 

For example, if Mathew has \(x\) pencils and John has 2 pencils more than Mathew, then John has \(x + 2\) pencils.

\(x + 2\) is a polynomial.

One boy holds x number of pencils and another boy holds two more pencils than the other boy.

The above two polynomials \(x\) and \(x + 2\) contain one and two terms respectively.

What about polynomials that contain three terms? How would they be referred to?

These types of polynomials are called trinomials or trinomial polynomial.

Let's learn about trinomials in this session.

Lesson Plan

What Is a Trinomial?

Before learning about trinomials let's first understand what is a polynomial.

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 polynomial is a type of polynomial that contains only three terms.

terms of polynomials

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 examples of trinomials 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.


How to Factor Trinomial?

A trinomial can be factorized in many ways. 

Let's discuss each case.

Quadratic Trinomial in One Variable

The general form of quadratic trinomial formula in one variable is

\[ax^2 + bx + c\]

Where \(a, b, c\) are constant terms and neither \(a, b, or c\) is zero.

For the value of \(a, b, c\), if \(b^2 - 4ac > 0\), then we can always factorize a quadratic trinomial.

It means that \(ax^2 + bx + c = a(x + h)(x + k)\), where \(h\) and \(k\) are real numbers.

Now let's learn how to factorize a quadratic trinomial with an example.

Example

Factorize: \(3x^2 - 4x - 4\)

Solution

Step 1:- First multiply the coefficient of \(x^2\) and the constant term.

\[3 \times -4 = -12\]

Step 2:- Break the middle term \((-4x)\) such that on multiplying the resulting numbers, we get the result \(-12\) (obtained from the first step).

\[-4x = -6x + 2x\]

\[-6 \times 2 = -12\]

Step 3:- Rewrite the main equation by applying the change in the middle term.

\[3x^2 - 4x - 4 \\[0.2cm]
3x^2 - 6x + 2x - 4 \]

Step 4:- Combine the first two terms and the last two terms, simplify the equation and take out any common numbers or expressions.

\[3x^2 - 6x + 2x - 4 \\[0.2cm]
3x(x - 2) + 2(x - 2) \]

Step 5:- Again take \((x - 2)\) common from both the terms.

\[3x(x - 2) + 2(x - 2) \\[0.2cm]
(x - 2)(3x + 2) \]

\((x-2)\) and \((3x + 2)\) are the factors of \(3x^2 - 4x - 4\)

Quadratic Trinomial in Two Variable

There is no specific way to solve a quadratic trinomial in two variables.

Let's take an example.

Example

Factorize: \(x^2 + 3xy + 2y^2\)

Solution

Step 1:- These type of trinomials also follow the same rule as above, i.e., we need to break the middle term.

\[x^2 + 3xy + 2y^2\\[0.2cm]
x^2 + 2xy + xy + 2y^2\]

Step 2:- Simplify the equation and take out common numbers of expressions.

\[x^2 + 2xy + xy + 2y^2 \\[0.2cm]
x(x + 2y) + y(x + 2y)\]

Step 5:- Again take \((x + 2y)\) common from both the terms.

\[x(x + 2y) + y(x + 2y) \\[0.2cm]
(x + y)(x + 2y)\]

\((x + y)\) and \((x + 2y)\) are the factors of \(x^2 + 3xy + 2y^2\)

If Trinomial is an Identity

Let's see some algebraic identities.

Identity Expanded Forms
\((x + y)^{2}\) \(x^2 + 2xy + y^2\)
\((x - y)^{2}\) \(x^2 - 2xy + y^2\)
\((x^2 - y^2)\) \((x + y)(x - y)\)

Example

Factorize: \(9x^2 + 12xy + 4y^2\)

Solution

Step 1:- Identify which identity can be applied in the expression.

We can apply \((x + y)^{2} = x^2 + 2xy + y^2\)

Step 2:- Rearrange the expression so that it can appear in the form of the above identity.

\[9x^2 + 12xy + 4y^2\]

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

Step 3:- Once the expression is arranged in the form of the identity, write its factors.

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

\[{(3x + 2y)}^{2}\]

\[(3x + 2y)(3x + 2y)\]

\((3x + 2y)\) is the factor of \(9x^2 + 12xy + 4y^2\)

Explore the trinomial calculator to factorize any trinomial.

 
important notes to remember
Important Notes
  • The degree of a polynomial is the highest power of the variable.
  • The polynomial with more than three terms is simply called a polynomial.
  • If the value of a quadratic trinomial with a single variable is zero, then it is known as a quadratic equation.
    \[ax^2 + bx + c = 0\]
  • If \((x - a)\) is a factor of \(\text{P(x)}\), then \(\text{P(a) = 0}\).

Solved Examples

Example 1

 

 

Help Tim find the factors of \(x^2 - 5x + 6\).

Solution

\[x^2 - 5x + 6 \\[0.2cm]
= x^2 - 3x - 2x + 3 \times 2 \\[0.2cm]
= x(x - 3) - 2x + 6 \\[0.2cm]
= x(x - 3) - 2(x - 3) \\[0.2cm]
= (x - 2)(x - 3)\]

\(\therefore\) Answer is \(x^2 - 5x + 6 = (x - 2)(x - 3)\)

Example 2

 

 

Help Clara find the factors of \(15a^2 + 38ab + 24b^2\)

Solution

Here we need to break the middle term. 

First, multiply the coefficient of \(a^2\) and \(b^2\)

\(15 \times 24 = 360\)

Now find two numbers such that on multiplication they give the result 360 and on addition, they give the result 38. 

\(18 \times 20 = 360\) and \( 18 + 20 = 38\) 

\[15a^2 + 38ab + 24b^2 \\[0.2cm]
= 15a^2 + 18ab + 20ab + 24b^2 \\[0.2cm]
= 3a(5a + 6b) + 4b(5a + 6b) \\[0.2cm]
= (5a + 6b)(3a + 4b) \]

\(\therefore\) \(15a^2 + 38ab + 24b^2 \!=\! (5a + 6b)(3a + 4b)\)
Example 3

 

 

If \(y - 3\) is a factor of \(y^2 + a - 6y\), then find the value of \(a\).

Find the other factor of the trinomial.

Solution

\(y - 3\) is a factor of \(y^2 + a - 6y\).

Then if we put \(y = 3\) in the trinomial \(y^2 + a - 6y\), its value will be 0.

\[\begin{align}3^2 + a - 6\times3 &= 0 \\[0.2cm]
9 + a - 18 &= 0 \\[0.2cm]
a - 9 &= 0 \\[0.2cm]
a &= 9\end{align}\]

Now factorize the trinomial \(y^2 + a - 6y = y^2 + 9 - 6y\)

The above trinomial is the expansion of the identity \({(x - y)}^{2} = x^2 - 2xy + y^2\)

\[y^2 - 6y + 9 \\[0.2cm]
= y^2 - 2\times3\times y + 3^2 \\[0.2cm]
= {(y - 3)}^{2} \]

\(\therefore\) \(a = 9\) and \(y^2 + a - 6y = {(y - 3)}^{2}\)
 
Challenge your math skills
Challenging Questions
  • Factorize: \(a^2 - b^2 + 12b - 36\)
  • Factorize: \(x^2 - y^2 + 2x + 1\)

Interactive Questions on Trinomial

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 

Let's Summarize

The mini-lesson targeted in the fascinating concept of trinomials. The math journey around trinomials starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

About Cuemath

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Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we at Cuemath believe in.


Frequently Asked Questions (FAQs)

1. What is a perfect square trinomial?

Perfect square trinomials are the trinomials which follows the identity \({(a + b)}^{2} = a^2 + 2ab + b^2\)

For example, \(x^2 + 2x + 1\) and \(4y^2 - 20y + 25\).

2. What is a cubic trinomial?

A cubic trinomial is a trinomial which has degree 3

For example, \(x^3 + 2x + 4\) and \( 2y^3 - 3y^2 + 1\).

3. What is a quadratic trinomial?

A quadratic trinomial is a polynomial with three terms and the degree of the trinomial must be 2

It means that the highest power of the variable cannot be greater than 2

For example: \(x^2 + y^2 + xy\) and \(x^2 + 2x + 3xy\).

It does not mean that a quadratic trinomial always turns into a quadratic equation when we equate it to zero.

A quadratic equation is a quadratic trinomial formula with only one variable.

You can factorize a trinomial using a trinomial calculator.

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