Chapter 1 : Introduction to Polynomial
Topics covered in this snacksized chapter:
Polynomials are mathematical expressions consisting of a sum of variables and exponents.
2x^{2
}+ 4x + 3
Polynomials DO NOT contain:
 Square roots of variables
 No variables in the denominator of any fraction
Let n be a nonnegative integer.
A polynomial function of degree n in the variable x is a function defined by:
is called the leading coefficient.
is called the dominating term.
is called the constant term.
is the yintercept of the graph of .
P(x) = 5x^{3
}+ 3x + 7
5 is the leading coefficient.
5x^{3
} is the dominating term.
The degree of this polynomial is 3 (Exponent of the leading term).
7 is the constant.
P(0) = 7 is the yintercept of the graph of P.
Polynomial graphs are smooth and continuous.
The graph shown below represents a polynomial function f(x) = x^{2
}.
Type of Graph
 Properties

Smooth Graph
 Rounded curves with no sharp corners.

Continuous Graph
 Graph has no breaks and can be drawn without lifting the pencil from the rectangular coordinate system.

Like Terms
 Unlike terms

The like terms have same variable and same degree.
 The unlike terms have different variables and different degrees.

2x and 4x are the like terms.
 3x^{2
}and 4y^{3
}are the unlike terms.

The degree of a polynomial is the largest exponent in the polynomial.
 When the variable does not have an exponent, it is assumed to be 1.
Examples:
 3x is the first degree polynomial.
 3x^{2
}is the second degree polynomial. 3x^{5
} 4x + 3 is the fifth degree polynomial.
The coefficient is the constant term placed before a variable.
In terms 3x and ax, 3 and a are coefficients of x.
Determine coefficient of each term ?
Solution:
Term
 Coefficient

x^{3
}
 1

7x^{2
}
 7

ax
 a

Type
 Examples

Monomial:
Has one term
 5y or – 8x

Binomial:
Has two terms
 3x^{2
} + 2
or
9y – 2y^{2
}

Trinomial:
Has three terms
 3x^{2
}+ 3x + 2
or
9y^{3
} – 2y^{2
} + y

Classification of Polynomials Based on Degree:
Degree
 Name
 Example


 Zero
 0

0
 (Nonzero) constant
 1

1
 Linear
 a + 1

2
 Quadratic
 a^{2
}+ 1

3
 Cubic
 a^{3
}+ 1

4
 Quartic(or biquadratic)
 a^{4
}+ 1

5
 Quintic
 a^{5
}+ 1

6
 Hexic
 a^{6
}+ 1

7
 Septic
 a^{7
}+ 1

8
 Octic
 a^{8
}+ 1

9
 Nonic
 a^{9
}+ 1

10
 Decic
 a^{10
}+ 1

Algebraic expressions that consist of terms in the form of ax^{n
}.
 n is a nonnegative integer.
 a is called the coefficient of a term.
Example 1: 4x.
 Here we have one term with variable x and the exponent (or degree) as 1.
Example 2: 3x^{2
} + 2.
 Here we have two terms, with one variable x having the exponent (or degree) as 2 and another is constant value 2.
Example 1:
 This is a polynomial with one variable.
Example 2:
 This is a polynomial with one variable.
Polynomials in two variables are algebraic expressions consisting of terms in the form:
The degree of the polynomial is the largest sum of the exponents of two variables.
Degree of the above polynomial is
12 + 22 = 34
Place like terms together.
Add the like terms.
Solve (5x + 7y) + (2x  y)
(5x + 7y) + (2x  y)
Group like terms together:
(5x + 2x) + (7y  y)
On simplifying the terms, we get:
= (7x + 6y)
To subtract polynomials, first reverse the sign of each term you are subtracting.
In other words turn "+" into "", and "" into "+" and then add as usual.
Solve (4x^{2
} 4)  (x^{2
} 4x + 4)
First arrange the terms:
 4x^{2
}– 4 + x^{2
}+ 4x  4
 = 4x^{2
}+ x^{2
} + 4x – 4 – 4
To multiply two polynomials:
 Multiply each term in one polynomial by each term in the other polynomial.
 Add the resulting terms together, and simplify if needed.
(x + 3)(x) + (x + 3)(2)
First arrange the terms as:
= (x + 3)(x) + (x + 3)(2)
= x(x) + 3(x) + x(2) + 3(2)
= x^{2
}+ 3x + 2x + 6
= x^{2
}+ 5x + 6
If f(x) and d(x) are polynomials,
 With d(x) = divisor and f(x) = dividend
 And the degree of d(x) is less than or equal to the degree of f(x).
 Then there exist unique polynomials q(x) and r(x) such that:
Arrange the terms of both the dividend and the divisor in descending powers of any variable.
 Divide the first term in the dividend by the first term in the divisor.
 The result is the first term of the quotient.
Multiply every term in the divisor by the first term in the quotient.
 Write the resulting product beneath the dividend with like terms lined up.
Subtract the product from the dividend.
 Bring down the next term in the original dividend and write it next to the remainder to form a new dividend.
Use this new expression as the dividend and repeat this process until the remainder can no longer be divided.
This will occur when the degree of the remainder (the highest exponent on a variable in the remainder) is less than the degree of the divisor.
Divide by .
First, set up the division.
Ignore the other terms and look just at the leading x of the divisor and the leading of the dividend.
Divide by the leading.
Put on top.
Take that x, and multiply it through the divisor, x + 1.
 Multiply the x (on top) by the x (on the “side”), and carry the underneath.
 Then multiply the x (on top) by the 1 (on the “side”), and carry the 1x underneath.
 To subtract the polynomials, change all the signs in the second line and then add down.
 The first term () will cancel out,
 Now, carry down that last term 10,
Now look at the x from the divisor and the new leading term, –10x, in the bottom line of the division.
 Divide the –10x by the x, it ends up with –10, so put that on top:
 Now multiply –10 by the leading x:
 Now multiply the –10 (on top) by the 1 (on the “side”), and carry the –10 to the bottom:
Change the signs of all the terms in the bottom row (to subtract).
Finally add down
The solution to this division is: x  10
L.H.S. =
= R.H.S.
Hence proved L.H.S = R.H.S