__Algebraic Expressions__

An algebraic expression is an expression made up of variables and constants along with mathematical operators.

An algebraic expression is a sum of terms, which are considered to be building blocks for expressions.

A term is a product of variables and constants. A term can be an algebraic expression in itself.

Examples of a term – 3 which is just a constant.

– 2x, which is the product of constant ‘2’ and the variable ‘x’

– 4xy, which is the product of the constant ‘4’ and the variables ‘x’ and ‘y’.

– 5x^{2}y, which is the product of 5, x, x and y.

The constant in each term is referred to as the coefficient.

Example of an algebraic expression: 3x^{2}y+4xy+5x+6 which is the sum of four terms: 3x2y, 4xy, 5x and 6.

An algebraic expression can have **any number of terms**. The **coefficient** in each term can be **any real number**. There can be **any number of variables** in an algebraic expression. The **exponent** on the variables, however, must be **rational numbers.**

**Polynomial**

An algebraic expression can have exponents that are rational numbers. However, a polynomial is an algebraic expression in which the exponent on any variable is a whole number.

5x^{3}+3x+1 is an example of a polynomial. It is an algebraic expression as well.

2x+3√x is an algebraic expression, but not a polynomial. – since the exponent on x is 1/2 which is not a whole number.

**Degree of a Polynomial**

For a polynomial in one variable – the highest exponent on the variable in a polynomial is the degree of the polynomial.

Example: The degree of the polynomial x^{2}+2x+3 is 2, as the highest power of x in the given expression is x2.

**Types Of Polynomials**

Polynomials can be classified based on:

a) Number of terms

b) Degree of the polynomial.

Types of polynomials based on the number of terms

a) Monomial – A polynomial with just one term. Example: 2x, 6x^{2}, 9xy

b) Binomial – A polynomial with two terms. Example: 4x^{2}+x, 5x+4

a) Trinomial – A polynomial with three terms. Example: x^{2}+3x+4

Types of Polynomials based on Degree

**Linear Polynomial**

A polynomial whose degree is one is called a linear polynomial.

For example, 2x+1 is a linear polynomial.

**Quadratic Polynomial**

A polynomial of degree two is called a quadratic polynomial.

For example, 3x^{2}+8x+5 is a quadratic polynomial.

**Cubic Polynomial**

A polynomial of degree three is called a ** cubic polynomial**.

For example, 2x

^{3}+5x

^{2}+9x+15 is a cubic polynomial.

**Graphical Representations**

Let us learn here how to represent polynomial equation on the graph.

Representing Equations on a Graph

Any equation can be represented as a graph on the Cartesian plane, where each point on the graph represents the x and y coordinates of the point that satisfies the equation. An equation can be seen as a constraint placed on the x and y coordinates of a point, and any point that satisfies that constraint will lie on the curve

For example, the equation y = x, on a graph, will be a straight line that joins all the points which have their x coordinate equal to their y coordinate. Example – (1,1), (2,2) and so on.

Geometrical Representation of a Linear Polynomial

The graph of a linear polynomial is a straight line. It cuts the X-axis at exactly one point.

**Linear graph**

Geometrical Representation of a Quadratic Polynomial

The graph of a quadratic polynomial is a parabola

It looks like a U which either opens upwards or opens downwards depending on the value of ‘a’ in ax^{2}+bx+c

If ‘a’ is positive, then parabola opens upwards and if ‘a’ is negative then it opens downwards

It can cut the x-axis at 0, 1 or two points

Graph of a polynomial which cuts the x-axis in two distinct points (a>0)

Graph of a Quadratic polynomial which touches the x-axis at one point (a>0)

Graph of a Quadratic polynomial that doesn’t touch the x-axis (a<0)

Graph of the polynomial x^n

For a polynomial of the form y=x^{n} where n is a whole number:

as n increases, the graph becomes steeper or draws closer to the Y-axis

If n is odd, the graph lies in the first and third quadrants

If n is even, the graph lies in the first and second quadrants

The graph of y=−x^{n} is the reflection of the graph of y=x^{n} on the x-axis

Graph of polynomials with different degrees.

**Zeroes of a Polynomial**

A zero of a polynomial **p(x)** is the value of x for which the value of p(x) is 0. If k is a zero of p(x), then **p(k)=0.**

For example, consider a polynomial p(x)=x^{2}−3x+2.

When x=1, the value of p(x) will be equal to

p(1)=12−3×1+2

=1−3+2

=0

Since p(x)=0 at x=1, we say that 1 is a zero of the polynomial x^{2}−3x+2

**Geometrical Meaning of Zeros of a Polynomial**

Geometrically, zeros of a polynomial are the points where its graph cuts the x-axis.

(i) One zero (ii) Two zeros (iii) Three zeros

Here A, B and C correspond to the zeros of the polynomial represented by the graphs.

**Number of Zeros**

In general, a polynomial of degree n has at most n zeros.

- A linear polynomial has one zero,
- A quadratic polynomial has at most two zeros.
- A cubic polynomial has at most 3 zeros.

**Factorisation of Polynomials**

Quadratic polynomials can be factorized by splitting the middle term.

For example, consider the polynomial 2x^{2}−5x+3

**Splitting the middle term:**

The middle term in the polynomial 2x^{2}−5x+3 is -5x. This must be expressed as a sum of two terms such that the product of their coefficients is equal to the product of 2 and 3 (coefficient of x^{2} and the constant term)

−5 can be expressed as (−2)+(−3), as −2×−3=6=2×3

Thus, 2x^{2}−5x+3=2x^{2}−2x−3x+3

Now, identify the common factors in individual groups

2x^{2}−2x−3x+3=2x(x−1)−3(x−1)

Taking (x−1) as the common factor, this can be expressed as:

2x(x−1)−3(x−1)=(x−1)(2x−3)

**Relationship between Zeroes and Coefficients of a Polynomial**

**For Quadratic Polynomial:**

If α and β are the roots of a quadratic polynomial ax^{2}+bx+c, then,

α + β = -b/a

Sum of zeroes = -coefficient of x /coefficient of x^{2}

αβ = c/a

Product of zeroes = constant term / coefficient of x^{2}

**For Cubic Polynomial**

If α,β and γ are the roots of a cubic polynomial ax^{3}+bx^{2}+cx+d, then

α+β+γ = -b/a

αβ +βγ +γα = c/a

αβγ = -d/a

**Division Algorithm**

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.

**Algebraic Identities**

1. (a+b)^{2}=a^{2}+2ab+b^{2}

2. (a−b)^{2}=a^{2}−2ab+b^{2}

3. (x+a)(x+b)=x^{2}+(a+b)x+ab

4. a^{2}−b^{2}=(a+b)(a−b)

5. a^{3}−b^{3}=(a−b)(a^{2}+ab+b^{2})

6. a^{3}+b^{3}=(a+b)(a^{2}−ab+b^{2})

7. (a+b)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3}

8. (a−b)^{3}=a^{3}−3a^{2}b+3ab^{2}−b^{3}