Squaring a Trinomial requires multiplying the trinomial by itself using the distributive property and then simplifying the expression by combining like terms. This process is fundamental in algebra and provides an expanded form of the trinomial.
In this article, we will learn about the Trinomial Definition, How to Square Trinomial, and Different Methods of Squaring a Trinomial with some solved examples to understand better.
What is a Trinomial?
A trinomial refers to a mathematical expression or equation that consists of three terms. These terms are algebraic expressions or variables combined using addition and subtraction operations. Trinomials can take various forms, such as quadratic trinomials, where the highest power of the variable is squared, or cubic trinomials, where the highest power is cubed.
General form of a trinomial is often expressed as ax2 + bx + c, where a, b, and c represent coefficients, and x is the variable. An example of a trinomial is,
Trinomial
Examples of Trinomials
Some examples of trinomials are:
- 2x2 + 5x – 3
- -4y2 – 2y + 7
- 3a2 – 6a + 1
How to Square a Trinomial
Squaring a trinomial involves multiplying the trinomial by itself. The process follows the general pattern of the distributive property and is often used in algebraic manipulations or solving mathematical equations. To square a trinomial in the form (ax2 + bx + c), you would multiply it by itself using the distributive property and then simplify the resulting expression.
For example squaring the trinomial (x2 + 2x + 3)
= (x2 + 2x + 3)2
Using Distributive Property:
= (x2 + 2x + 3)(x2 + 2x + 3)
= x2(x2 + 2x + 3) + 2x(x2 + 2x + 3) + 3(x2 + 2x + 3)
= x4 + 2x3 + 3x2 + 2x3 + 4x2 + 6x + 3x2 + 6x + 9
Simplifying and combining like terms,
= x4 + 4x3 + 10x2 + 12x + 9
This process can be applied to any trinomial by following the same steps of multiplying each term in the trinomial by every term in the trinomial and then simplifying the result.
We can also use the Squaring a Trinomial Formula to find the square of a trinomial.
Squaring a Trinomial Formula
Squaring a Trinomial Formula is,
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2ac + 2bc
Also,
(ax2 + bx + c)2 = a2x4 + 2abx3 + (2ac + b2)x2 + 2bcx + c2
Related, Perfect Square Trinomials Formula
Methods of Squaring a Trinomial
There are few methods of squaring a trinomial are:
- Distributive Property Method
- Binomial Expansion Method
Distributive Property Method
(ax2 + bx + c)2
Applying distributive property:
(ax2 + bx + c)(ax2 + bx + c)
Distributing each term in first trinomial to every term in the second trinomial:
a(ax2) + a(bx) + ac + b(ax2) + b(bx) + bc + c(ax2) + c(bx) + c2
Simplifying and combining like terms:
a2x4 + 2abx3 + (2ac + b2)x2 + 2bcx + c2
Binomial Expansion Method
Another method involves using the binomial expansion formula i.e.
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2ac + 2bc
How to Expand Square of a Trinomial?
Expanding the square of a trinomial involves multiplying the trinomial by itself and simplifying the resulting expression. Let’s use a general trinomial (ax2 + bx + c) as an example and go through the steps to expand its square (ax2 + bx + c)2
Step 1: Apply Distributive Property
(ax2 + bx + c)2
= (ax2 + bx + c)(ax2 + bx + c)
Distribute each term in the first trinomial to every term in the second trinomial:
= a(ax2) + a(bx) + ac + b(ax2) + b(bx) + bc + c(ax2) + c(bx) + c2
Step 2: Simplify and Combine Like Terms
= a2x4 + 2abx3 + (2ac + b2)x2 + 2bcx + c2
Combine like terms to simplify the expression.
So, expanded form of (ax2 + bx + c)2 is [a2x4 + 2abx3 + (2ac + b2)x2 + 2bcx + c2]
Also Read,
Examples of Squaring a Trinomial
Some examples on squaring a trinomial are,
Example 1: Given trinomial 3x2 – 2x + 5, find square of this trinomial.
Solution:
Given trinomial,
(3x2 – 2x + 5)2
Applying Distributive Property:
(3x2 – 2x + 5)(3x2 – 2x + 5)
Distributing each term
9x4 – 6x3 + 15x2 – 6x3 + 4x2 – 10x + 15x2 – 10x + 25
Combining like terms
9x4 – 12x3 + 24x2 – 20x + 25
So, square of trinomial (3x2 – 2x + 5) is (9x4 – 12x3 + 24x2 – 20x + 25)
Example 2: A rectangular garden has an area represented by the trinomial expression 2x2+7x-4. If the length of the garden is represented by (2x + 1) units, find the width of the garden.
Solution:
Area of a Rectangle is,
Area = Length × Width
In this case, area of rectangular garden is represented by trinomial expression (2x2 + 7x – 4), and the length is given as (2x + 1) units.
Let the width be represented by (w). So, we have:
2x2 + 7x – 4 = (2x) × w
Now, we can solve for (w):
2x2 + 7x – 4 = 2xw
Divide both sides by (2x):
w = (2x2 + 7x – 4)/2x
Now, simplify the expression
w = (2x + 1)(x – 4)/(2x + 1)
w = (x – 4)
So, width of garden is given by (x – 4)
Example 3: Calculate Square of Trinomial -4a2 + 3a – 1
Solution:
= (-4a2 + 3a – 1)2
Using Distributive Property
= (-4a2 + 3a – 1)(-4a2 + 3a – 1)
= 16a4 – 12a3 + 4a2 – 12a3 + 9a2 – 3a + 4a2 – 3a + 1
= 16a4 – 24a3 + 13a2 – 6a + 1
So, square of trinomial (-4a2 + 3a – 1) is (16a4 – 24a3 + 13a2 – 6a + 1)
Practice Questions on Squaring a Trinomial
Various practice questions on squaring a trinomial are,
Q1. The area of a square field is given by the trinomial x2+6x+9. Determine the side length of the square field.
Q2. If p2-5p+4 represents the square of a binomial, find the possible values of p.
Q3. The volume of a cube is represented by the trinomial 4x2-12x+9. Determine the length of each side of the cube.
Q4. Find the square of the trinomial 2y2+7y-3.
Q5. The area of a rectangular room is given by the trinomial 3x2+8x-5. If the length of the room is 3x+5 meters, find the width of the room.
Frequently Asked Questions-Squaring a Trinomial
What is a Trinomial in Math?
Trinomial in math are type of polynomial that contains only three terms. For example, x2 + 3x – 11, etc.
What are Perfect Square Trinomial?
A perfect square trinomial is a trinomial that can be factored into the square of a binomial. General form of a perfect square trinomial is a2 + 2ab + b2
What is a Trinomial Square?
A trianomial square is (ax2 + bx + c)2 = a2x4 + 2abx3 + (2ac + b2)x2 + 2bcx + c2
How do you Square a Trinomial?
A trinomial is squared using the methods of distributive property and binomial expansion
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