UNIT 2: POLYNOMIAL EQUATIONS &
INEQUALITIES
Synthetic Division of Polynomials
Factoring x^n-y^n
2.1 The Remainder Theorem (pg.84)
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The remainder of a polynomial 𝑓(𝑥) dividing by 𝑥 − 𝑎 is 𝑓(𝑎)
𝑓(𝑥)
𝑟(𝑥)
○
= 𝑞(𝑥) + 𝑥 − 𝑎
𝑥−𝑎
𝑓(𝑥) = 𝑑(𝑥)𝑞(𝑥) + 𝑟(𝑥) or 𝑓(𝑥) = (𝑥 − 𝑎)𝑞(𝑥) + 𝑟(𝑥)
■ 𝑓(𝑥); dividend
■ 𝑑(𝑥); divisor = 𝑥 − 𝑎
■ 𝑞(𝑥); quotient
■ 𝑟(𝑥); remainder = 𝑐, a constant one degree less than divisor
○ 𝑓(𝑎) = 𝑐
} If the remainder is 0, that tells us the divisor is a FACTOR of the dividend
𝑏
The remainder of a polynomial 𝑓(𝑥) dividing by 𝑎𝑥 − 𝑏; where 𝑎 ≠ 0 is 𝑓( 𝑎 )
○
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Solving for Unknown Coefficient
○ Given the remainder and polynomial with an unknown variable
○ Solve for k when 𝑓(𝑎) = 𝑟𝑒𝑚𝑎𝑖𝑛𝑑𝑒𝑟
- Synthetic Division
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Using only the coefficients of the terms
2.2 The Factor Theorem (pg.94)
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Using the remainder theorem
○ The polynomial 𝑓(𝑥) has a factor of 𝑥 − 𝑝 if and only if 𝑓(𝑝) = 0
■ If 𝑓(𝑝) = 0, 𝑓(𝑥) = (𝑥 − 𝑝)𝑞(𝑥); 𝑥 − 𝑝 is a factor
■ If 𝑥 − 𝑝 is a factor, 𝑟 must be 0; 𝑓(𝑝) = 0
○ if the remainder = 0; the binomial is a FACTOR of the polynomial
Integral Zero Theorem
○ The relationship between the factors of a polynomial and the constants
○ Test all ± factors of 𝑟 (the remainder)
■ If you find a factor insert 𝑥 − 𝑝 and divide
Rational Root Theorem
○
𝑛
𝑛−1
The polynomial 𝑓(𝑥) = 𝑎𝑛𝑥 + 𝑥
where 𝑞 ≠ 0
○
𝑎
𝑟
Test all ± factors of 𝑎0 or 𝑎
𝑛
■
𝑝
+... 𝑎1 + 𝑎0 has a factor 𝑞𝑥 − 𝑝 if and only if 𝑓( 𝑞 ) = 0;
𝑛
If you find a factor insert 𝑞𝑥 − 𝑝 and divide
2.3 Polynomial Equations
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Solutions / Roots
○ 𝑓(𝑥) = 0
○ The zeros and/or x-int of the function 𝑦 = 𝑓(𝑥)
Solving Polynomial Functions
○ For {XER}, a function of degree 𝑛 has UP TO 𝑛 (REAL) roots
The Fundamental Theorem of Algebra
○ Every polynomial 𝑃(𝑧) of degree 𝑛 has EXACTLY 𝑛 values of 𝑧 for which 𝑃(𝑧 ) = 0
𝑖
𝑖
Where 𝑧 is a complex number
Coefficients of 𝑃 and {ZEℂ}; where 𝑧 = 𝑎 + 𝑏𝑖
■
○
𝑎 } real
𝑏𝑖 } imaginary (but still real)
■
■
○
2
𝑖 =
−1
𝑖 =
|
𝑛
−1
𝑛
- Binomial Theorem (Factoring 𝑥 − 𝑎 )
2
2
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Diff of squares; 𝑥 − 𝑎
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Pattern; 𝑥 − 𝑎
𝑛
𝑛
3
= (𝑥 − 𝑎)(𝑥 − 𝑎) |
𝑛−1
= (𝑥 − 𝑎)(𝑥
𝑛−2
+ 𝑎𝑥
3
Diff of cubes; 𝑥 ± 𝑎
2 𝑛−3
+𝑎 𝑥
𝑛−2
+ . . . 𝑥𝑎
2
𝑛−1
+𝑎
)
𝑛
○
𝑥 ; where 𝑛 decrease until you reach 0x
○
𝑎 ; where 𝑛 increase until you reach 𝑛 − 1
𝑛
2.4 Families of Polynomial Functions
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Family of Functions
○ Set of functions that have the same characteristics
○ The graph of the polynomial functions have same x-intercepts but different y-intercepts
■ With the exception; if one of the x-intercepts are zero
○ 𝑦 = 𝑘(𝑥 − 𝑎 )(𝑥 − 𝑎 )(𝑥 − 𝑎 )... (𝑥 − 𝑎 ); 𝑘 ≠ 0
1
2
3
𝑛
2.5 & 2.6 Polynomial Inequalities
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To solve an inequality using a graph, important to know the location of the zeros/𝑥-intercepts
○ The function MAY change signs at these 𝑥-values
Solving Algebraically
○ Factor the inequality; determine the zeros
○ Using one of the following 3 methods to determine the interval or set notation
■ Method 1: Number Line
■ Method 2: Factor Table
● Order 𝑥-𝑖𝑛𝑡 from left to right
■
x < #1
#1 < x < #2
x > #2
Factor that obtained #1
-
+
+
Factor that obtained #2
-
-
+
f
+
-
+
If the table is order correctly, should create the triangles for + and - values
Method 3: Sketch Graph
● Use degree of the polynomial
● Use Leading Coefficient
● Using Zeros
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2
= (𝑥 ± 𝑎)(𝑥 ∓ 𝑎𝑥 + 𝑎 )