Algebra:

a) Algebraic Functions:

In algebra, we have to deal with variables, constants or a mixture of both.

All those expressions having at least one variable are called algebraic expressions.

Example: ax+by+3, mx+6, etc.

Similarly, all those equations having the algebraic terms are called algebraic equations. Eg: y= mx + 5, y = 2x, etc.

These are some of the algebraic Functions:

Linear Function:

Remember a formula that we studied in the previous classes: y = mx+c? If you do, then, we know that it represents the equation of a straight line having slope and an intercept. That was in coordinate geometry.

Here too, any algebraic function in the form of y=mx+c, where m and c are constants and the equation is of first degree is called linear equation.

Above definition means to say that, the variables should have power of 0 or 1. Meaning, the values should be constant and should not have the power of two, three or more.

Constant Function:

A constant function is such type of algebraic function that has the equation of the form f(x) = c, where c is a constant.

Here, this means to say that the function returns the same output regardless of the input given.

This is a type of linear equation because when m= 0 in the equation y = mx +c, we obtain y= c.

Remember, that 'c' should not be a variable! It must be constant to satisfy constant function.

'y= c'; this can also be written as f(x) = c.

Let's take 'c' as 2 and a constant function f(x) =2 with the domain {1,2,3,....n}

Our function would be:

f(1) =2, f(2) =2, f(3) =2,..... f(n)=2

Identity Function:

An identity function is such type of algebraic function which returns the same value of the constant used as its argument.

Here, this means to say the the input and the output of the function is the same value. If we input value '1', your output would be '1'.

Identity means being almost similar.

This is called as a linear function because it is derived from the equation of linear function, y= mx+c. When c = 0 and m= 1 in the linear equation, we get;

y = x

This can be written as, f(x) = x

Let's take an identity function whose functional value is defined by f(x) = x.

Let the domain of the function be {1,2,3,...,n}

We get, f(1) =1, f(2) =2, f(3) =3,....., f(n) = n

b) Quadratic Function:

A Quadratic Function is such type of algebraic function that has the equation in the form of y=ax²+bx+c, where a is number other than zero and b and c are any numbers.

Here, this means the degree of this function is 2. If the highest exponent in any function is equal to x², then the given function is quadratic function.

When you draw the graph of a quadratic function, you get a graph with a smooth curve. Hence, we receive parabola and vertex.

Parabola is the smooth curve obtained by the graph of quadratic equation.

Vertex is the turning point of the parabola in the graph of quadratic equation.

Let us take an example that, a=1, and b=c=0,

Our function becomes; f(x) = x²

When we have, domain of the function = {-1,0,1}

We get;

f(-1) = 1

f(0) = 0

f(1) = 1

So, f={(-1,1),(0,0),(1,1)}

Drawing this function in a graph, we get:

c) Cubic Function:

A Cubic Function is such type of algebraic function that has the equation in the form of y= ax³+bx²+cx+d, where a is any number other than zero and b,c and d are any numbers.

Here, this means the degree of this function is 3. If the highest exponent in any function is equal to x³, then the given function is cubic function.

Let us take an example of the cubic function where a =1, and b=c=d=0, we have; f(x) = x³.

Let us take an example that the domain of the range = {-1,0,1)

We get,

f(-1) = -1

f(0) = 0

f(1) = 1

So, f= {(-1,-1),(0,0),(1,1)}

Trigonometric Function:

Important notes:

The y-intercept of the function f(x) = sinx is 0.
The y-intercept of the function f(x) = cosx is 1.
The y-intercept of the function f(x) = tanx is 0.

Trigonometric function is such type of function that deals with the angles, expressed as the ratio of the sides of a right angled triangle.

For example: f(x) = sin x, f(a) = cosA +6, etc.

To make it easier, if in any function, we need to use the Trigonometric values of the angles then, such type of function is said to be Trigonometric function.

Learn More about Trigonometry to understand more about Trigonometric Functions.

Understanding:

Domain:

The trigonometric angle that we input to any function is said to be the domain of that function.

Range:

The value that we obtain from the Trigonometric function is said to be its range.

For example: f(x) = sin30°

30° is the domain and the value 0.5 is the range.

Division of Polynomials:

Click here to learn more about Polynomials.

Like in arithmetic division, we have divisor, dividend, quotient and remainder. When we have to divide 5 by 2, we have 5 as dividend, 2 as divisor, 2 as quotient and 1 as remainder.

Similarly, the polynomials which is to be divided is known as dividend, polynomial that divides the dividend is divisor, the polynomial that we obtain during division is quotient, and the remaining polynomial that we obtain is remainder.

Things to remember:

Divisor should not be equal to zero.

Remainder may or may not be equal to zero.

Let, f(x) and g(x) be two polynomials. Q(x) be the quotient, f(x) be the divisor, g(x) be the dividend and r(x) be the remainder.

We get;

g(x) = Q(x). f(x) + r(x) is called division algorithm.

Synthetic Division:

The quickest, fastest and the easiest way of polynomial division is synthetic division. It is used to divide a polynomial f(x) by a binomial g(x)=(x-a).

Here are the steps of synthetic division:

1. Compare (x-2) with (x-a). [We get a = 2.]

2. Now, write the leading coefficients of the given algebraic terms. [As in above example; we have: 1,-6,11,-6.]

3. Write the same coefficient of the highest degree below. [In above example, 1.]

4. Then multiply the coefficient you obtained in step 3 by a. [We have, 1*2 = 2.]

5. Now, add the result you get in step 4 with the next coefficient obtained in step 2. [ Here, -6+2 = -4.]

6. Go-to step 4.

7. The results that you obtain will have the following bases: x², x, and none. [Here, 1x² -4x +3]

This is how you perform syntethic division.

When you do not have the binomial as (x-a) rather (px-q).

Then, a = q/p

Then, actual quotient = Q(x)/p

Remainder Theorem:

Remainder Theorem states that, "Let p(x) be any polynomial of degree n, n>=1, and g(x) = (x-a), where a is any number. If p(x) is divided by g(x), then the remainder is p(a).

Basically, to find the remainder while dividing a polynomial by a binomial (x-a), we have the formula;

Remainder = p(a)

Note:

When the binomial is (x+a), then the remainder is p(-a).

When the binomial is (ax+b), then the remainder is p(-b/a).

When the binomial is (ax-b), then the remainder is p(b/a).

View all solutions of Remainder Theorem.

Factor Theorem :

Factor Theorem states that, "Let p(x) be any polynomial of degree n, n>=1, is divided by a linear polynomial (x-a) and if remainder p(a) = 0, where a is a real number, then (x-a) is a factor of p(x).

Basically, it means if the remainder is zero, the given divisor is the factor of the dividend.

View Solutions related to Factor Theorem.

Polynomial Equation:

Polynomial Equation is an equation consisting of constants and variables.

Let p(x) be a polynomial that is equal to 0, the value of x which satisfies the relation p(x) = 0 is said to be a root.

The exact value that you obtain by substituting the value of x in p(x) is called the value of polynomial.

The value of x, which when substituted in p(x), reduces the value of polynomial p(x) to zero is called zero of polynomial.

Learn here more about Sequence and Series

Arithmetic Sequence and Series:

Arithmetic sequence are those sequence having equal difference between the consecutive terms.

In Arithmetic Sequence (AS),

When the first term = a,

Common difference between consecutive terms = d

The nth term or general term of the AS is

tn = a + (n-1) d

Arithmetic Mean:

The average of the given numbers obtained by adding all given number and dividing them by the number of terms is said to be arithmetic mean.

Important Formulae:

Mean of two Numbers

= (a+b)/2

Arithmetic Mean between two Numbers = difference between two terms = (b-a)/(n+1)

In Arithmetic Sequence,

First mean (m1) = Second term (t2) = first term + difference

or, m1 = t2 = a + (b-a)/(n+1)

Second mean (m2) = Third term (t3) = first term + 2*difference

or, m2 = t3 = a + 2(b-a)/(n+1)

Third mean (m3) = Fourth term (t4) = first term + 3*difference

or, m3 = t4 = a + 3(b-a)/(n+1)

So, m_n = a + nd = a + n (b-a)/(n+1)

Sum of the Arithmetic Series:

Here, we have the formulae to get the sum of the Arithmetic Series in the following two conditions:

1) The first term is 'a' and last term is 'b'

Sn = n/ 2 (a+b)

2) The first term is 'a' and common difference is 'd'

Sn = n/2 [2a + (n-1)d]

Geometric Sequence and Series:

Geometric sequence are those sequence having same ratio of difference between the consecutive terms.

In Geometric Sequence,

When first term = a

Constant ratio = r

Then, tn = a r^(n-1)

We also have,

When first term = a

When last term = b

Constant ratio = r

Then, b = a r^(n-1) ... (i)

So, r = (b/a)^{1/(n-1)}

Geometric Mean:

The average of the given numbers obtained by adding all given number and dividing them by the number of terms is said to be arithmetic mean.

Important Formulae:

Geometric Mean between two known numbers:

= √ab

Geometric Mean between two numbers:

= a ( b/a )^{ n /( n+1)}

Relation between arithmetic mean and geometric mean:

Arithmetic Mean is always greater than or equal to Geometric mean between two positive real numbers.

Sum of Geometric Series:

Here, we have the formula to find the sum of Geometric Series in the following two conditions:

1) When a, r and n are known:

Sn = { a(r^n -1) / (r-1) } [See the image below for clarification ]

2) When a, r and b are known:

Sn = (br - a) / (r -1)

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Algebra - Unit 1 | Class 10 | Notes and Important Formulae | Optional Mathematics

Algebra:

Contents

a) Algebraic Functions: