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Polynomials

Polynomial
An algebraic expression in which the exponent of variable in each term is a non-negative integer is called a polynomials. A polynomial with variable x is denoted by f(x), g(x), h(x), etc. A polynomial with variable y is denoted by f(y), g(y), h(y) etc.

Degree of Polynomial
The highest degree of the given polynomial is known as its degree.
E.g. In x + 2, the degree is 1.
In 3x2 + 5x + 6, the degree is 2.
In x3 – 6x2 + 11x + 6, the degree is 3.

Types of Polynomials

  • Monomial
    A polynomial with only one term is called a monomial. E.g. 5x, 6y, 5xy etc.
  • Binomial
    A polynomial with two terms is called a binomial. E.g. 5x2 + 2x, 2x – 3, 5xy + 4, etc.
  • Trinomial
    A polynomial with three terms is known as a trinomial. E.g. x2 + 5x + 4, x2 + 9x + 14, etc.
  • Multinomial
    A polynomial with more than three terms is called multinomial. E.g. x4 + 2x3 + x2 + 3x + 2, y3 + 4y2 + 5y + 6 etc.

Synthetic Division
It is then divided by comparing with division with x-a and a becomes the divisor. Then the quotient is determined with a degree less than the dividend and the last number becomes the remainder.
Example: Divide g(x) = x2 + 5x + 6 by f(x) = x – 2
Solution:
Comparing   with x – a of f(x) then
x – 2 = 0
or x = 0 + 2
\(\therefore\) x = 2
Now,
\(\begin{array}{c|lcr} 2 & 1 & 5 & 6 \\   &   & 2 & 14 \\ \hline   & 1 & 7 & |20| \\ \end{array}\)
Hence,
Quotient = x + 7
Remainder = 20

Exercise

  1. Find the sum of f(x) = 2x + 3 and g(x) = 9x + 8
    Solution:
    Here,
    f(x) = 2x + 3
    g(x) = 9x + 8
    Now,
    f(x) = g(x)
    \(\begin{matrix} & 2x & + & 3 \\  & 9x & + & 8 \\ \hline (+) & 11x & + & 11 \end{matrix}\)

  2. Find the sum of f(x) = x3 + x2 + 4x + 8 and g(x) = x3 + 3x2 – 9x + 20
    Solution:
    Here,
    f(x) = x3 + x2 + 4x + 8
    g(x) = x3 + 3x2 – 9x + 20
    Now,
    f(x) + g(x) = \( \begin{matrix} & x^3 & + & x^2 & + & 4x & + & 8 \\ (+) & x^3 & + & 3x^2 & - & 9x & + & 20 \\ \hline & 2x^3 & + & 4x^2 & - & 5x & + & 28 \end{matrix}\)

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