द्विघाती बहुपद

द्विघाती बहुपद वह होता है जिसमें बहुपद व्यंजक में एक चर पद की उच्चतम घात $2$ के समान होता है। द्विघाती बहुपद को द्वितीय-क्रम बहुपद के रूप में भी जाना जाता है।

परिभाषा

द्विघाती बहुपद एक द्वितीय-घात बहुपद है जहां उच्चतम घात पद का मान $2$ के समान होता है। द्विघात समीकरण का सामान्य रूप $ax^{2}+bx+c=0$ के रूप में दिया जाता है। यहां, $a$ और $b$ गुणांक हैं, $x$ अज्ञात चर है और $c$ है स्थिर पद. चूँकि इस समीकरण में एक द्विघाती बहुपद है, अतः इसे हल करने पर दो समाधान मिलेंगे। इसका तात्पर्य यह है कि $x$ के दो मान हो सकते हैं।

उदाहरण

$x^{2}+4x+4=0$

इस समीकरण का हल खोजने के लिए हम इसका गुणनखंड इस प्रकार करते हैं

$x^{2}+4x+4=0$

$x^{2}+2x+2x+4$

$x(x+2)+2(x+2)$

$(x+2)(x+2)=0$

इस प्रकार इस द्विघाती समीकरण के मूल $x=-2,x=-2$ होंगे

Quadratic Polynomial Formula

The general formula of a single variable quadratic polynomial is given as $ax^{2}+bx+c$. When this quadratic polynomial is used in an equation it is expressed as $ax^{2}+bx+c=0$. There are many methods that can be used to find the solutions of an equation containing a quadratic polynomial. These methods are factorizing a quadratic equation, completing the squares, using graphs, and using the quadratic polynomial formula. Out of all these techniques, the simplest way to find the roots of a quadratic polynomial is by using the formula. An added benefit of this method is that several important conclusions can be made by analyzing the discriminant. The quadratic polynomial formula is given below:

$x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}$

The two values of $x$ that are obtained after applying this formula are known as the solutions, the zeros or the roots of the quadratic equation.

The value $b^{2}-4ac$ is called the discriminant. It is denoted by $D$.The nature of the roots can be determined by using the discriminant.

Quadratic Polynomial Roots

The method of factorization is only applicable to certain quadratic polynomials. However, the quadratic polynomial formula can be used for any type of quadratic equation. Furthermore, the value of the discriminant can be used to analyze the nature of the roots of a quadratic polynomial. Given below are the various conditions that can help to predict the nature of the roots:

• $D>0$: If the discriminant is positive, it indicates that the roots are real and distinct.
• $D=0$: If the value of the discriminant is equal to zero, then both the roots are real and are equal to each other.
• $D<0$: Both the roots are imaginary numbers if the discriminant is negative.

Quadratic Polynomial Sum and Product of Roots

Using the roots of the equation containing the quadratic polynomial, a relationship can be established between the roots and the coefficients. The sum and the product of the roots of a quadratic polynomial can be determined using the coefficients and the constant term. Suppose one root is given by $\alpha$ and the other root is given by $\beta$

. For a quadratic equation, $ax^{2}+bx+c=0$, containing a quadratic polynomial, the formula for the sum and product of roots is given below:

• Sum of roots: $\alpha +\beta =-$Coefficient of $x$ / Coefficient of $x^{2}=-{\frac {b}{a}}$
• Product of roots: $\alpha .\beta =$ Constant / Coefficient of $x^{2}={\frac {c}{a}}$

If the sum and product of the roots has been specified then the original quadratic polynomial can be obtained. This is given by

$x^{2}-(\alpha +\beta )x+\alpha .\beta =0$

This can also be used to factor qudratic polynomials. Other methods for factorizing a quadratic polynomial will be listed in sections below.

How to Find Quadratic Polynomial?

A quadratic polynomial can be obtained by using the zeros or roots of the equation. Suppose the two roots are given as$-4$ and $2$. The steps to find the quadratic polynomial are as follows:

• Step 1: Find the sum of the two roots. Sum of roots$=-4+2=-2$
• Step 2: Find the product of the two roots. Product of roots$=-4\times 2=-8$
• Step 3: Substitute these values in the expression $x^{2}-$ (sum of the roots)$x+$(product of the roots). Thus, the quadratic polynomial is $x^{2}+2x-8$

How to Factorize Quadratic Polynomials?

Generally, factorization can be considered as the reverse of multiplying two expressions. Few methods for factorization of quadratic polynomials are listed below:

Common Factor Method

In this method, we have to look at all the terms and determine the common terms.

If there is a common term in the equation, we will factor it out for the polynomial.

We use distributive law in reverse.

$x(a+b)=xa+xb$

We notice that each term has an '$x$' in the equation and the common factor is taken out using the distributive law in reverse as follows,

$xa+xb=x(a+b)$

Example

What are the common factors of the terms in the quadratic polynomial equation $8x^{2}-4x=0?$

Solution

Let's apply the distributive law in reverse.

$4x$ is a common factor in the equation.

Thus, $4x(2x-1)$are the factors of$8x^{2}-4x=0$

Sum of Difference Method

The sum and the difference of two terms are most likely used when the two factors match exactly, except one term involves addition and the other is a difference.

For example: $(a+b)(a-b)$

When we expand and multiply these terms, we get $a\times a+ab-ab-b\times b$

Like terms will be in the middle and will result in zero, thus leaving behind $a^{2}$and $-b^{2}$

Thus, the formula becomes $(a+b)(a-b)=a^{2}-b^{2}$

Example

Find the solution of $(5+x)(5-x)$ using the sum of the difference method.

Solution

Apply the sum of the difference method for solving the terms.

$(a+b)(a-b)=a^{2}-b^{2}$

$(5+x)(5-x)=(5^{2}-x^{2})=25-x^{2}$

Factor By Grouping Method

Factor by grouping means that we have to group all the terms with common factors before factoring.

The following steps are used in the factor by grouping method.

• From the given quadratic polynomial, take out a factor from each group.
• Factorize each group of the expression.
• Now take out the factor common to the group formed.

Let’s take a look at an example.

Example

How can you factorize the quadratic polynomial $a^{2}-ac+ab-bc$ by the grouping method?

Solution:

$a^{2}-ac+ab-bc$

Take the common factor from the quadratic polynomial.

$=a(a-c)+b(a-c)$

$=(a-c)(a+b)$

Thus, by factoring expressions we get $(a-c)(a+b)$

Perfect Square Trinomials Method

The method of converting any quadratic polynomial into a perfect square is known as the perfect square trinomial method.

The following equations are the perfect square trinomial formulas:

$a^{2}+2ab+b^{2}=(a+b)^{2}$

$a^{2}-2ab+b^{2}=(a-b)^{2}$

Example

Is the given quadratic polynomial $x^{2}-8x+16$ a perfect square?

Solution

On using the formula, we get

$x^{2}-8x+16=x^{2}-2(1)(4)x+4^{2}=(x-4)^{2}$

Thus, the given quadratic polynomial is a perfect square.