Section 3.2: Polynomial Division
Lecture 11
– Typeset by FoilTEX –
Outline
30
• Long Division of Polynomials
20
10
• Remainders and Factors
-2
1
-1
-10
• Factors and Zeros
f HxL = Hx + 2L Hx + 1L Hx - 1L Hx - 2L Hx - 3L
-20
-30
2
3
Polynomial Division
• The degree of a polynomial is its largest exponent.
∗ The degree of f (x) = x2 + x + 1 is 2.
∗ The degree of g(x) = x + x6 − x3 − 11 is 6.
∗ The degree of h(x) = 3 is 0.
• Given two polynomials f (x) and p(x), we could divide f (x) by p(x).
• We would find two polynomials, a quotient q(x) and a remainder
r(x), such that
f (x) = p(x)q(x) + r(x)
and the degree of r(x) is less than the degree of p(x).
Example
Find the quotient q(x) and the remainder r(x) if
f (x) = 3x4 + 2x3 − x2 − x − 6
is divided by
p(x) = x2 + 1.
Example
Find the quotient q(x) and the remainder r(x) if
f (x) = x6 + x5 − x − 1
is divided by
p(x) = x5 − 1.
Remainder and Factor Theorems
• Remainder Theorem: When a polynomial f (x) is divided by
p(x) = x − c, the remainder is r(x) = f (c).
(Easy to find the remainder in this case.)
∗ f (x) = q(x)(x − c) + r(x)
∗ f (c) = q(c) · 0 + r(c)
• Factor Theorem: For a polynomial f (x), x − c is a factor of f (x) if
and only if f (c) = 0.
• This is an x-intercept finder!
Example
• Show that c = 2 is a zero of f (x) = −12 + 4x + 15x2 − 5x3 − 3x4 + x5.
• Divide f (x) by x − 2 (call the quotient q1(x)).
• Show that c = 1 is a zero of q1(x).
• Divide q1(x) by x − 1 (call the quotient q2(x)).
• Show that c = −1 is a zero of q2(x).
• Divide q2(x) by x + 1
• Factor f (x) completely.