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**Introduction to Polynomial Equation**

If p(x) is a polynomial equation in x, then the highest power of x in p(x) is called the degree of the polynomial p(x). So, p(x) =

4x + 2 is a polynomial equation in the variable x of degree 1

2y^{2} â€“ 3y + 4 is a polynomial in the variable y of degree 2

5x^{3} â€“ 4x^{2} + x â€“ 2 is a polynomial in the variable x of degree 3

7u6 â€“ 3u4 + 4u2 â€“ 6 is a polynomial in the variable u of degree 6

Further, it is important to note that the following expressions are NOT polynomials:

- 1 / (x â€“ 1)
- âˆšx + 2
- 1 / (x
^{2}+ 2x + 3)

**Types of Polynomials**

Letâ€™s look at the different types of polynomials that you will come across while studying them.

Linear Polynomials

Any polynomial with a variable of degree one is a Linear Polynomial. Some examples of the linear polynomial equation are as follows:

- 2x â€“ 3
- y + âˆš2
- x âˆš3 + 5
- x + 5/11
- 2/3y â€“ 5

Any polynomial where the degree of the variable is greater than one is not a linear polynomial.

Quadratic Polynomials

Any polynomial with a variable of degree two is a Quadratic Polynomial. The name â€˜quadraticâ€™ is derived from the word â€˜quadrateâ€™ which means square. Some examples of the quadratic polynomial equation are as follows:

2x^{2} + 3x â€“ 5

y^{2} â€“ 1

2 â€“ x^{2} + xâˆš3

u/3 â€“ 2u^{2} + 5

v2âˆš5 + 2/3v â€“ 6

4z^{2} + 1/7

To generalize, most quadratic polynomials in x are expressed as, ax^{2} + bx + c â€¦ where a, b, and c are real numbers. Also, a â‰ 0.

Cubic Polynomials

Any polynomial with a variable of degree three is a Cubic Polynomial. Some examples of the cubic polynomial equation are as follows:

x^{3}

2 â€“ x^{3}

x^{3}âˆš2

x^{3} â€“ x^{2} + 3

3x^{3} â€“ 2x^{2} + x â€“ 1

To generalize, most quadratic polynomials in x are expressed as, ax^{3} + bx^{2} + cx + d â€¦ where a, b, c, and d are real numbers. Also, a â‰ 0.

**Some more concepts**

To begin with, letâ€™s look at the following polynomial p(x),

p(x) = x^{2} â€“ 3x â€“ 4

Next, letâ€™s put x = 2 in p(x). So, we get p(2) = (2)^{2} â€“ 3(2) â€“ 4 = 4 â€“ 6 â€“ 4 = â€“6. Note that the value â€˜â€“ 6â€™ is obtained by replacing x with 2 in the polynomial x2 â€“ 3x â€“ 4. Hence, it is called â€˜the value of x^{2} â€“ 3x â€“ 4 at x = 2â€™. Similarly, p(0) is the value of x^{2} â€“ 3x â€“ 4 at x = 0.

Therefore, we can say, If p(x) is a polynomial in x, and if k is any real number, then the value obtained by replacing x by k in p(x), is called the value of p(x) at x = k and is denoted by p(k).

Zero of a Polynomial

So, what is the value of p(x) = x^{2} â€“ 3x â€“ 4 at x = â€“ 1? p(â€“ 1) = (â€“ 1)2 â€“ {3(â€“ 1)} â€“ 4 = 1 + 3 â€“ 4 = 0. Also, the value of p(x) = x^{2} â€“ 3x â€“ 4 at x = 4 is, p(4) = (4)2 â€“ 3(4) â€“ 4 = 16 â€“ 12 â€“ 4 = 0. In this case, since p(â€“ 1) and p(4) is equal to zero, â€˜â€“ 1â€™ and â€˜4â€™ are called zeroes of the quadratic polynomial x^{2} â€“ 3x â€“ 4.

Therefore, we can say, A real number k is said to be a zero of a polynomial p(x) if p(k) = 0. In the previous years, you have already studied how to find zeroes of a polynomial equation. To elaborate on it a little more, if â€˜kâ€™ is a zero of p(x) = 2x + 3, then

p(k) = 0

Or 2k + 3 = 0

i.e. k = â€“ 3/2

Letâ€™s generalize this. If â€˜kâ€™ is a zero of p(x) = ax + b, then p(k) = ak + b = 0. Or, k = â€“ b/a. In other words, the zero of the linear polynomial (ax + b) is: â€“ (Constant Term) / (Coefficient of x)

Hence, we can conclude that the zero of a linear polynomial equation is related to its coefficients. You will study more about if this rule is applicable to all types of polynomials discussed above.

**Solved Examples for You**

**Question: **What are the three types of polynomials and how are they differentiated?

**Solution:** The three types of polynomials are:

- Linear
- Quadratic
- Cubic

The linear polynomials have a variable of degree one, quadratic polynomials have a variable with degree two and cubic polynomials have a variable with degree three.

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