Download Section 5.1: Polynomial Functions as Mathematical Models

Intermediate Algebra Test #03 Review Sheet
Friday 10/22/04
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Section 5.1: Polynomial Functions as Mathematical Models
1. Definition: A monomial is an algebraic expression that is either a constant or a product of constants and
one or more variables with whole number exponents.
2. Definition: A polynomial is a finite sum of monomials.
3. Definition: The degree of a monomial is the sum of the exponents of its variables. The degree of a
nonzero constant is zero.
4. Definition: The degree of a polynomial is the highest degree of any monomial in it.
5. Definition: A polynomial in one variable is a function of the form
P(x) = anxn + an-1xn-1 + … + a2x2 + a1x + a0. n is the degree of the polynomial.
Section 5.2: Polynomials: Sums, Differences, and Products
1. To add or subtract polynomials, combine like terms.
2. Distributive property: a(b + c) = ab + ac.
3. To multiply polynomials, use distribution, or multiply each term in the first polynomial by each term in
the second polynomial.
a. Example: (x2 + 3x + 9)(x2 – 2x +3) = x4 –2x3 +3x2
+3x3 -6x2 +9x
+9x2 -18x +27
4
3
=x
+x
+6x2 -9x
+27
Section 5.3: General Forms and Special Products
1. General Form #1: (x + a)(x + b) = x2 + (a + b)x + ab
2. General Form #2: (ax + b)(cx + d) = acx2 + (ad + bc)x + bd
3. FOIL Method: First, Outer, Inner, Last
a. Example: (2x + 3)(5x-9) = 10x2 – 18x + 15x – 27 = 10x2 – 3x – 27
4. Difference of Two Squares: (a + b)(a – b) = a2 – b2
5. Square of Sum: (a + b)2 = a2 + 2ab + b2
6. Square of Difference: (a – b)2 = a2 – 2ab + b2
Section 5.4: Factoring Out the Greatest Common Factor
1. Common Monomial Factoring: Factor out the greatest monomial (i.e., constants and variables of the
highest power possible) common to all terms in a polynomial.
a. Example: 3x2 – 15x4 + 81x6 = 3x2(1 – 5x2 + 27x4)
b. Example: 5a(x – 2y) – 3b(x – 2y) = (5a – 3b)(x – 2y)
2. Factoring by Grouping: Grouping and factoring parts of a polynomial to factor the polynomial itself.
a. Example: 3xb – 2b – 15x + 10 = 3xb – 2b – 15x + 10 = b(3x – 2) – 5(3x – 2) = (b – 5)(3x – 2)
Section 5.5: Factoring Trinomials
1. Factoring x2 + qx + p:
a. Use the fact that (x + a)(x + b) = x2 + (a + b)x + ab
b.  look for factors of p that sum to q
c. Example: x2 – 4x – 12 =
i. factors of 12 are: 1, 2, 3, 4, 6, 12
ii. the factors that add up to –4 are –6 and +2
iii.  x2 – 4x – 12 = (x – 6)(x + 2)
2. Factoring Ax2 + Bx + C:
Intermediate Algebra Test #03 Review Sheet
Friday 10/22/04
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a. Use the fact that (ax + b)(cx + d) = acx2 + (ad + bc)x + bd
b.  list factors of A and B, then try out all the possibilities until you get it right
c. Example: 2x2 – 9x – 18 =
i. factors of 2 are: 1, 2
ii. factors of 18 are: 1, 2, 3, 6, 9, 18
iii. (2x – 2)(x + 9) = 2x2 + 18x – 2x – 18 = 2x2 – 16x – 18
iv. (2x + 3)(x – 6) = 2x2 – 12x + 3x – 18 = 2x2 – 9x – 18 
3. Factoring Trinomials by Grouping:
a. You’re on your own…
4. Factoring Trinomials by the AC Method:
a. You’re on your own…
5. Factoring Using Special Products:
a. Difference of Two Squares: a2 – b2 = (a + b)(a – b)
i. Example: 9x2 – 81 = (3x + 3)(3x – 3)
b. Square of Sum: a2 + 2ab + b2 = (a + b)2
i. Example: x2 + 12x + 36 = (x + 6)2
c. Square of Difference: a2 – 2ab + b2 = (a – b)2
i. Example: x2 - 2x + 1 = (x – 1)2
d. Difference of Cubes: a3 – b3 = (a – b)(a2 + ab + b2)
i. Example: x3 – 8 = (x – 2)(x2 + 2x + 4)
e. Sum of Cubes: a3 + b3 = (a + b)(a2 – ab + b2)
i. Example: 27x3 + 64 = (3x + 4)(9x2 – 12x + 16)
Section 5.6: Solving Polynomial Equations by Factoring
1. Zero-Product Rule: If a  b = 0, then either a = 0 or b = 0.
2. To solve a (polynomial) nonlinear equation, get all terms on one side, then factor and apply the zero
product rule.
a. Example:
x3 – 4x2
= 12x
x3 – 4x2 – 12x = 0
x(x3 – 4x2 – 12)= 0
x(x – 6)(x + 2) = 0
x = 0 or
x–6=0
or
x+2=0
x=0
x=6
x = -2
Intermediate Algebra Test #03 Review Sheet
Friday 10/22/04
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Section 5.7: Polynomial Division
1. Example:
2 x3  x  18
?
x3
2 x 2  6 x  17
x  3 2 x 3  0 x 2  x  18

 2 x3  6 x 2

6x2  x

 6 x 2  18 x

 17 x  18
  17 x  51
33

2 x3  x  18
33
 2 x 2  6 x  17 
x3
x3
Section 6.1: Rational Functions
1. A rational function is a function of the form f ( x) 
p ( x)
, where p(x) and q(x) are polynomials and
q ( x)
q(x)  0.
Section 6.2: Equivalent Fractions
ak a
 (b, k  0)
bk b
x 2  y 2 ( x  y )( x  y ) ( x  y )
a. Example:


( x  y)2 ( x  y)( x  y) ( x  y)
1. The Fundamental Principle of Fractions:
Section 6.3: Multiplication and Division of Rational Expressions
a c ac
1. Multiplication of Fractions:  
(b, d  0)
b d bd
a c a d
2. Division of Fractions:    (b, c, d  0)
b d b c
x 3
( x  3)
4 2 x  1 4( x  3)(2 x  1)
a. Example:
 (8 x  4) 
 

4
2
2 x  5x  3
(2 x  1)( x  3) 1
1
( x  3)(2 x  1)
Intermediate Algebra Test #03 Review Sheet
Friday 10/22/04
Section 6.4: Sums and Differences of Rational Expressions
1. Addition and Subtraction of Rational Expressions:
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a b ab
 
(Note the need for a common
c c
c
denominator)
3x  1 2 x  1  3x  1 x  1  2 x  1 x  1



x 1 x 1
 x  1 x  1  x  1 x  1
2. Example: 

(3x 2  2 x  1)  (2 x 2  3x  1)
 x  1 x  1
x2  5x  2
 x  1 x  1
Section 6.5: Mixed Operations and Complex Fractions
Keys:
1. Remember order of operations
2. Multiply numerator and denominator by the L.C.D. of all fractions…
Section 6.6: Fractional Equations and Inequalities
Key: Multiply both sides of the equation (or inequality) by the L.C.D. of all fractions…
Section 6.7: Literal Equations
Key: Solve these types of equations the same as you would any other type of linear or nonlinear equation.
Section 6.8: Applications
See previous chapters’ notes on solving word problems.