Polynomial equations

An expression in the form of

f(x) = anxn + an-1xn-1 + … + a2x2 + a1x + ao

where n is a non-negative integer and a2, a1, and a0
are real numbers.

 The function is called a
polynomial function of x with
degree n.
 A polynomial is a monomial or a
sum of terms that are
monomials.
 Polynomials can NEVER have a
negative exponent or a variable
in the denominator!

Degree Name Example
0 Constant 5
1 Linear 3x+2
2 Quadratic X2 – 4
3 Cubic X3 + 3x + 1
4 Quartic -3x4 + 4
5 Quintic X5 + 5x4 - 7

MSTI - OTC SF 9/10/2012 4

 The graphs of polynomial functions are
continuous (no breaks—you draw the entire
graph without lifting your pencil).
 This is opposed to discontinuous functions
(remember piecewise functions?).
 This data is continuous as opposed to
discrete.

MSTI - OTC SF 9/10/2012 5

 The graph of a polynomial function has only
smooth turns. A function of degree n has at
most n – 1 turns.
◦ A 2nd degree polynomial has 1 turn
◦ A 3rd degree polynomial has 2 turns
◦ A 5th degree polynomial has…

MSTI - OTC SF 9/10/2012 6

 In arithmetic division we know that when we
divide one number by another there is, in
general, a quotient and a remainder.
 48 / 5
 Then 48 is called dividend
 5 is called divisor
 9 is called quotient
 3 is called remainder

 In the algebra of polynomials too a
polynomial f(x) can be divided by a
polynomial g(x) provided that the degree of
f(x) is greater than or equal to the degree of
g(x). Here f(x) is the dividend and g(x) is the
divisor. The quotient and remainder obtained
in this division are, in general, polynomials.

 If the degree of f(x) is n and the degree of
g(x) is m, then the degree of the quotient is
(n-m) and the degree of the remainder is at
most (m-1).

 If
◦ f(x)=anxn+an-1xn-1+an-2xn-2+…+a1x+a0, xЄ R and
◦ g(x)=bnxn+bn-1xn-1+bn-2xn-2+…+b1x+b0 , xЄ R

Consider
f(x)/g(x)
the degree of the quotient is (n-m) and the degree of
the remainder is at most (m-1).

1. Arrange both dividend and divisor in the
descending order.
2. Divide the first term of the dividend by the
first term of the divisor to obtain the first
term of the quotient.
3. Multiply the divisor by the term found in 2
above and subtract the result from the
dividend.
4. Annex to this remainder the unused terms of
the dividend to get a partial dividend.

5. Divide the first term of above by the first
term of the divisor to obtain the second term
of the quotient.
6. Multiply the divisor by the term found in 5
above and subtract.
7. Repeat this process until the remainder is 0
or the remainder is less than the degree of
the divisor.

When the divisor is of the form
x-a, the method of division
given above can be shortened

start by writing only the
coefficients of the dividend
and the divisor after arranging
them in the descending order

The dividend can be obtained by
adding the remainder to the
product of the divisor and quotient.

 Example

 If a polynomial f(x) is divided by x-a, the
remainder is f(a).

 Now if the remainder is zero, that is f(a)=0,
x-a is a factor of f(x) and, conversely, if x-a
is a factor of f(x) the remainder is zero. This
is called the factor theorem.

 This theorem can be used to factorize
polynomials of degree 3 and above.

 In order to find factors using the factor
theorem trial and error methods will have to
be used. For this purpose choose numbers
that are factors of the independent term. In
the above example substituting 2 or -2 is of
no use because they are not factors of 15.

Using long division find the quotient and remainder.
(1)
(2)
(3)
(4)
(5)

1. x3-5x2+x+16 is divided by x-2
2. X3+7x2-3x is divided by x+3
3. 2x4+3x3-5x+7 is divided by x+2
4. 3x4+x3-12x2-11x-24 is divided by x+2
5. 2x3-5x2+11x+6 is divided by 2x+1
6. 2x4+7x3-12x2+2x-5 is divided by 2x-1

1. Find the remainder when:
1. X3+3x2-4x+2 is divided by x-1
2. X3-x2+5x+8 is divided by x+2
3. 2X3+5x2+3x+11 is divided by x+2
4. X5+7x2-x+4 is divided by x+2
5. 4X3-2x2+x+7 is divided by 2x-1
6. 4X3+6x2+3x+2 is divided by 2x+3

Polynomial equations

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