2015-2016 Unit 5: Polynomials II
Algebra II CP
SUM and DIFFERENCE of CUBES
Sum of Two Cubes: a3  b3  (a  b)(a 2  ab  b 2 )
Difference of Two Cubes: a3  b3  (a  b)(a 2  ab  b 2 )
EXAMPLE: Polynomials that cannot be factored are called prime polynomials. Factor each polynomial. If the
polynomial cannot be factored, write prime.
(1.)
x3  y 3
(2.)
y 3  x3
(3.)
64 w3  125 z 3
(4.)
8 y 3  27 z 3
Lesko
Page 1 of 13
2015-2016 Unit 5: Polynomials II
Algebra II CP
EXAMPLE: COMPLETELY Factor each polynomial. If the polynomial cannot be factored, write prime.
(5.)
x3  5 x 2  2 x  10
(6.)
a 2  3ay  2ay 2  6 y 3
(7.)
3 x 4  12 x 2  15
(8.)
2 x 4  8 x 2  42
Lesko
Page 2 of 13
2015-2016 Unit 5: Polynomials II
Algebra II CP
(9.)
9 x 4  30 x 2  25
(10.)
x 4  16 x 2  64
(11.)
x4  y 4
(12.)
9 x 4  16 y 4
(13.)
x6  y 6
(14.)
x6  y 6
Lesko
Page 3 of 13
2015-2016 Unit 5: Polynomials II
Algebra II CP
EXAMPLE: SOLVE the polynomials (FIND THE ZEROS – X INTERCEPTS – ROOTS)
(15.)
18 x 4  21x 2  3  0
(16.)
4 x4  8x2  3  0
(17.)
8 x 4  10 x 2  12  0
(18.)
x 4  29 x 2  100  0
Lesko
Page 4 of 13
2015-2016 Unit 5: Polynomials II
REMAINDER THEOREM
Algebra II CP
If a polynomial P ( x ) is divided by x  r , the remainder
is a constant P ( r ) , where Q ( x) is a polynomial with degree
one less than P ( x ) .
EXAMPLE: Find the remainder.
(19.)
f ( x)  3x3  6 x 2  x  11 , find f (3)
(20.)
f ( x)  4 x5  2 x3  x 2  1 , find f ( 1)
EXAMPLE: Determine whether the binomial is a factor. If the factor is a remainder, then find the remaining.
(21.)
Lesko
Determine whether x  5 is a factor of
x3  7 x 2  7 x  15 .
(22.)
Determine whether x  2 is a factor of
x3  7 x 2  4 x  12 .
Page 5 of 13
2015-2016 Unit 5: Polynomials II
(23.)
Determine whether x  3 is a factor of
x3  4 x 2  15 x  18 .
FUNDAMENTAL THEOREM OF ALGEBRA
Algebra II CP
(24.)
Determine whether x  1 is a factor of
2 x3  17 x 2  23x  42 .
Every polynomial equation with a degree greater than zero
has at least one root in the set of complex numbers.
*Note: Real Numbers and Imaginary Numbers both
belong to the Complex Numbers *
EXAMPLE: Solve each equation. State the number and type of roots.
(25.)
Lesko
x2  6x  9  0
(26.)
x3  25 x  0
Page 6 of 13
2015-2016 Unit 5: Polynomials II
Algebra II CP
(27.)
x 4  16  0
(28.)
x5  8 x3  16 x  0
(29.)
y 4  256  0
(30.)
3x3  x 2  9 x  3  0
A polynomial equation of degree n has exactly n
roots in the set of complex numbers, including
repeated roots.
Descartes’ Rule of Signs
Corollary to the Fundamental Thereon of Algebra
The number of positive real zeros of P(x) is the same as the number of changes in
sign of the coefficients of the terms, or is less than this by an even number, and
The number of negative real zeros of P(x) is the same as the number of changes in
sign of the coefficients of the terms of P(-x), or is less than this by an even number.
Lesko
Page 7 of 13
2015-2016 Unit 5: Polynomials II
Algebra II CP
EXAMPLE: State the possible number of positive real zeros, negative real zeros, and imaginary zeros.
(31.)
f ( x)  x 6  3 x 5  4 x 4  6 x 3  x 2  8 x  5
(32.)
f ( x)  2 x 5  x 4  3x 3  4 x 2  x  9
(33.)
f ( x)   x 6  4 x 3  2 x 2  x  1
(34.)
f ( x)  x 4  2 x 2  5 x  19
Lesko
Page 8 of 13
2015-2016 Unit 5: Polynomials II
COMPLEX CONJUGATES
THEOREM
Algebra II CP
Let a and b be real numbers, and b cannot be zero. If a  bi is a zero of a
polynomial function with real coefficients, then a  bi is also a zero of the
function.
EXAMPLE: Write a polynomial function.
(35.)
The zeros include -1, -6, -8
(37.)
The zeros include 4 and 4  i
Lesko
(36.)
The zeros include -2, 5, -7
Page 9 of 13
2015-2016 Unit 5: Polynomials II
Algebra II CP
Rational Zero Theorem
If P(x) is a polynomial function with integral coefficients, then every rational zero of P(x) = 0 is of the form
p
,
q
a rational number in simplest form, where p is a factor of the constant term and q is a factor of the leading
coefficient term.
Corollary to the Rational Zero Theorem
If P(x) is a polynomial function with integral coefficients, a leading coefficient of 1, and a nonzero constant
term, then any rational zeros of P(x) must be factors of the constant term.
EXAMPLE: List all the possible rational zeros of the function.
(38.)
f ( x)  4 x5  x 4  2 x3  5x 2  8x  16
(39.)
f ( x)  x3  3x 2  5x  12
(40.)
f ( x)  3x3  4 x  10
(41.)
f ( x)  x3  11x 2  24
Lesko
Page 10 of 13
2015-2016 Unit 5: Polynomials II
Algebra II CP
EXAMPLE: Find all of the zeros.
(42.)
Lesko
f ( x)  5 x 4  8 x3  41x 2  72 x  36
Page 11 of 13
2015-2016 Unit 5: Polynomials II
(43.)
Lesko
Algebra II CP
f ( x)  2 x 4  5x3  20 x 2  45x  18
Page 12 of 13
2015-2016 Unit 5: Polynomials II
(44.)
Algebra II CP
Graph the function p  x    x  2 x  1 x  1 using its zeros.
80
70
60
50
40
30
20
10
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
-10
-20
-30
-40
-50
-60
-70
(a) Even or Odd:
(b) Start and End Behavior:
(b) x-intercepts:
(d) Number of Turning Points:
(e) y-intercepts:
(f) Local minima:
(g) Local maxima:
Lesko
Page 13 of 13