1. More about Polynomials
Chapter 6
More about Polynomials

2. Polynomials
A monomial is a number, or a product of a number and one or more variables with non-negative integral exponents.
axn is a monomial where n is a non-negative integer, a is a constant, and x is a variable.
3, 2x3, 4x2, … are monomials.
A polynomial is a sum of monomials e.g. x + 3, 3x2 - 4x + 5, x3 - 2x2 + 4x + 1 are polynomials.

3. Polynomials -5x3 - 2x2 + 4x - 7 Constant term = -7
Degree of polynomial = 3
-5x3 - 2x2 + 4x - 7
Coefficient of x3
= -5
Coefficient of x2
= -2
Coefficient of x
= 4

4. Like Terms and Unlike Terms
2x, -3x, 9x are called like terms.
2x, 3, 9x2 are called unlike terms.
Like terms can be added or subtracted to form a new term, but unlike terms cannot.
e.g. 2x - 3x + 9x = 8x

5. Fundamental Operations of Polynomials
(7x2 - 2x + 4) - (2x2 + 3x - 2)
= 5x2 - 5x + 6
(3x + 4)(x - 2)
= 3x(x - 2) + 4(x - 2)
= 3x2 - 6x + 4x - 8
= 3x2 - 2x - 8

6. Division of Polynomials (I)
Quotient
Divisor
Dividend
-5x + 4
Remainder 9
3x2 - 2x + 4 = (x + 1)(3x - 5) + 9

7. Division of Polynomials (II)1

8. Division of Polynomials (II)
Quotient +1
x + 1
Divisor
3x3 + 0x2 - 2x + 1
Dividend
-3x 2 - 2x
x + 1
Remainder
3x3 - 2x + 1 = (x + 1)(3x2 - 3x + 1)

9. Remainder Theorem
When a polynomial f(x) is divided by a linear polynomial x - a, the remainder R = f(a).
When a polynomial f(x) is divided by a linear polynomial mx - n, the
remainder

10. Factor Theorem
If a linear polynomial mx - n is a factor of polynomial f(x), i.e. f(x) is divisible by
mx - n, then
Conversely, if , then the linear
polynomial mx - n is a factor of f(x), i.e. f(x) is divisible by mx - n.

11. End