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**Polynomial **

An algebraic expression of the form p(x) = a_{0} + a_{1} x + a_{2} x^{2} + a_{3} x^{3} + ...... + a_{n} x^{n}, in which the variables involved have only **non-negative integral exponents**, is called a polynomial in x of degree n.

Note:In the polynomial a

_{0}+ a_{1}x + a_{2}x^{2}+ .... + a_{n}x^{n}1. a

_{n}≠ 02. a

_{0}, a_{1}x^{1}, a_{2}x^{2}, a_{3}x^{3}, ..... a_{n - 1}x^{n - 1}, a_{n}x^{n}are terms.3. a

_{0}, a_{1}, a_{2}, ..... a_{n - 1}, a_{n}are the co-efficients of x^{0}, x^{1}, x^{2}, ....., x^{n-1}, x^{n}respectively.

**Degree of a Polynomial **

The **highest power of the variable** in a polynomial is called its **degree**.

**Example:** 5x + 3 is a polynomial in x of degree 1.

p(y) = 3y^{2} + 4y - 4 is a polynomial in y of degree 2.

**Linear Polynomial:**A polynomial of**degree 1**is called a**linear polynomial**. A linear polynomial is generally written as ax + b (a ≠ 0), where a, b are real coefficients.**Quadratic Polynomial:**A polynomial of**degree 2**is called a**quadratic polynomial**. A quadratic polynomial is generally written as ax^{2}+ bx + c (a ≠ 0), where a, b and c are real coefficients.**Cubic Polynomial:**A polynomial of**degree 3**is called a**cubic polynomial**. A cubic polynomial is generally written as ax^{3}+ bx^{2}+ cx + d (a ≠ 0), where a, b, c and d are real coefficients.

**Examples:**

Try yourself:The degree of the polynomial, x^{4} – x^{2} +2 is

View Solution

**Value of a Polynomial at a Given Point **

If p (x) is a polynomial in x and ‘a’ is a real number. Then the value obtained by putting x = a in p (x) is called the value of p (x) at x = a.

**Example:** Let p(x) = 5x^{2} - 4x + 2 then its value at x = 2 is given by

p(2) = 5 (2)^{2} - 4 (2) + 2 = 5 (4) - 8 + 2 = 20 - 8 + 2 = 14

Thus, the value of p(x) at x = 2 is 14.

**Zeroes of a Polynomial**

A real number ‘a’ is said to be a zero of the polynomial p (x), if p (a) = 0.

**Example: **Let p (x) = x^{2} - x - 2 Then p (2) = (2)^{2} - (2) - 2 = 4 - 4 = 0,

and p (-1) = (-1)^{2} - (-1) - 2 = 2 - 2 = 0

∴ (-1) and (2) are the zeroes of the polynomial x^{2 }- x - 2.

Note:

I.A linear polynomial has at the most one zero.II.A quadratic polynomial has at the most two zeroes.

III.In general a polynomial of degree n has at the most n zeroes.

**Geometrical Meaning of the Zeroes of a Polynomial**

First, we consider a linear polynomial p (x) = ax + b.

Let ‘k’ be a zero, then p(k) = ak + b = 0

⇒ ak + b = 0

⇒ ak = - b or

The graph of a linear polynomial is always a straight line. It may or may not pass through the x-axis. In case the graph line is passing through a point on the x-axis, then the y-coordinate of that point must be zero. In general, for a linear polynomial ax + b = 0, (a ≠ 0), the graph is a straight line that can intersect the x-axis at exactly one point, namely, is the zero of the polynomial ax + b.

In the given figure, CD is meeting x-axis at x = -1.

∴ Zero of ax + b is -1.

Note:A zero of a linear polynomial is the x-coordinate of the point, where the graph intersects the x-axis.

**Graph of a Quadratic Polynomial**

The graph of ax^{2} + bx + c, (a ≠0) is a curve of ∪ shape, called a parabola.

- If a > 0 in ax
^{2}+ bx + c, the shape of the parabola is ∪ (opening upwards). - If a < 0 in ax
^{2}+ bx + c, the shape of parabola is ∩ (opening downwards).

In the given figure, the graph of a quadratic polynomial x^{2} - 3x - 4 is shown. It intersects x-axis at (-1, 0) and (4, 0). Therefore, its zeroes are -1 and 4. Here, a > 0, so the graph opens upwards.Whereas the following figure is a graph of the polynomial - x^{2} + x + 6. Since it intersects the x-axis at (3, 0) and (-2, 0). Therefore, the zeroes of - x^{2} + x + 6 are -2 and 3.

Here a < 0, so the parabola opens downwards.

Note:In the case of Quadratic polynomial - at most 2 zeroes, Cubic polynomial - at most 3 zeroes, Biquadratic polynomial - at most 4 zeroes.

**Relationship between Zeroes and Coefficients of Polynomials**

- For a quadratic polynomial,

p(x) = ax^{2}+ bx + c, α + β = -b/a, αβ = c/a,

where α and β are the zeroes of the polynomial p(x) = ax^{2}+ bx + c - For a cubic polynomial,

p(x) = ax^{3}+ bx^{2}+ cx + d,

where α, β and γ are the zeroes of the polynomial p(x) = ax^{3}+ bx^{2}+ cx + d.

Try yourself:What is the quadratic polynomial whose sum and the product of zeroes is √2, ⅓ respectively?

View Solution

If we divide a polynomial p(x) by a polynomial g(x), then their exists polynomials q(x) and r(x) such that, p(x) = g(x) x q(x) + r(x), where r(x) = 0 or deg r(x) < deg g(x).

To divide one polynomial by another, follow the steps given below.

**Step 1:** Arrange the terms of the dividend and the divisor in the decreasing order of their degrees.

**Step 2:** To obtain the first term of the quotient, divide the highest degree term of the dividend by the highest degree term of the divisor Then carry out the division process.

**Step 3:** The remainder from the previous division becomes the dividend for the next step. Repeat this process until the degree of the remainder is less than the degree of the divisor.

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