त्रिपद

A Trinomial is an algebraic expression that has three terms. An algebraic expression consists of variables and constants of one or more terms. These expressions use symbols or operations as separators such as $+,-,\times$ and $\div$. A trinomial along with monomial, binomial, and polynomial are categorized under this algebraic expression.

Definition

A Trinomial is an algebraic expression that has three non-zero terms and has more than one variable in the expression. A trinomial is a type of polynomial but with three terms. A polynomial is an algebraic expression that has one or more terms and is written as

$a_{0}x^{n}+a_{1}x^{n-1}+a_{2}x^{n-2}+.....+a_{n}x^{0}$ in the standard form, where $a_{0},a_{1},a_{2},.....a_{n}$ are constants and $n$ is a natural number.

Examples of trinomial with multiple variables and three terms are $x^{2}+y^{2}+xy$ , $xyz^{3}+x^{2}z^{2}+zy^{3}$

Examples of trinomials with one variable are $x^{2}+2x+3$ , $5x^{4}-4x^{2}+1$

A polynomial can be referred to by different names depending on the number of terms it has. The table below mentions the names.

Number of terms	Polynomial	Example
1	Monomial	$xy$
2	Binomial	$x+y$
3	Trinomial	$x^{2}+xz+1$

Quadratic Trinomial

A quadratic trinomial is a type of algebraic expression with variables and constants. It is expressed in the form of $ax^{2}+bx+c$, where $x$ is the variable and $a,b,c$ are non-zero real numbers. The constant '$a$' is known as a leading coefficient, '$b$' is the linear coefficient, '$c$' is the additive constant. A quadratic trinomial also describes the discriminant $D$ where it defines the quantity of an expression and it is written as $D=b^{2}-4ac$.The discriminant helps in classifying among the different cases of quadratic trinomials. If the value of a quadratic trinomial with a single variable is zero, then it is known as a quadratic equation i.e $ax^{2}+bx+c=0$

How to Factor Trinomials?

Factoring a trinomial means expanding an equation into the product of two or more binomials/monomials. It is written as $(x+m)(x+n)$.

A trinomial can be factorized in many ways. .

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 $a,b,c$ 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 $h,k$ are real numbers.

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 coefficient 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.

$3x^{2}-4x-4=3x^{2}-6x+2x-4$

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

$3x^{2}-6x+2x-4=3x(x-2)+2(x-2)$

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

$3x(x-2)+2(x-2)=(x-2)(3x+2)$

Therefore, $(x-2)$ 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.

Example: Factorize: $x^{2}+3xy+2y^{2}$

Solution:

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

$x^{2}+3xy+2y^{2}=x^{2}+2xy+xy+2y^{2}$

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

$x^{2}+2xy+xy+2y^{2}=x(x+2y)+y(x+2y)$

Step 3: Again take $(x+2y)$ common from both the terms.

$x(x+2y)+y(x+2y)=(x+y)(x+2y)$

Therefore, $(x+y)$ and $(x+2y)$ are the factors of $x^{2}+3xy+2y^{2}$ त्रिपद एक बीजगणितीय व्यंजक है जिसमें तीन पद होते हैं। बीजगणितीय व्यंजक में एक या अधिक पदों के चर और अचर होते हैं। ये व्यंजक +, -, ×, और ÷ जैसे प्रतीकों या संक्रियाओं को विभाजक के रूप में उपयोग करते हैं। इस बीजगणितीय व्यंजक के अंतर्गत एक त्रिपद के साथ-साथ एकपद, द्विपद और बहुपद को वर्गीकृत किया गया है।

परिभाषा

त्रिपद एक बीजगणितीय व्यंजक है जिसमें तीन गैर-शून्य पद होते हैं और व्यंजक में एक से अधिक चर होते हैं। त्रिपद एक प्रकार का बहुपद है लेकिन इसमें तीन पद होते हैं। बहुपद एक बीजगणितीय व्यंजक है जिसमें एक या अधिक पद होते हैं और इसे मानक रूप में $a_{0}x^{n}+a_{1}x^{n-1}+a_{2}x^{n-2}+.....+a_{n}x^{0}$ के रूप में लिखा जाता है, जहां $a_{0},a_{1},a_{2},.....a_{n}$स्थिरांक हैं और $n$ एक प्राकृतिक संख्या है।

अनेक चरों और तीन पदों वाले त्रिपदों के उदाहरण $x^{2}+y^{2}+xy$ , $xyz^{3}+x^{2}z^{2}+zy^{3}$हैं

एक चर वाले त्रिपदों के उदाहरण $x^{2}+2x+3$, $5x^{4}-4x^{2}+1$ हैं