๐Ÿƒ Flashcards ยท Topic 1.5

Polynomial Functions
& Complex Zeros

10 cards ยท 2 print sheets. Print double-sided, cut along dashed lines.

๐Ÿ“‹ Notes ๐Ÿƒ Flashcards ๐Ÿง  Quiz
Print double-sided (flip on long edge) โ†’ cut along dashed lines.
10 cards ยท 2 print sheets
๐Ÿ“„ Page 1 โ€” Questions FRONT ยท Sheet 1/2
1/10
Topic 1.5 ยท Multiplicity โ€” Odd
What does ODD multiplicity mean for a zero's graph behavior?
Crosses or bounces?
2/10
Topic 1.5 ยท Multiplicity โ€” Even
What does EVEN multiplicity mean for a zero's graph behavior?
Crosses or bounces?
3/10
Topic 1.5 ยท Conjugate Pairs
If a + bi is a zero of a polynomial with real coefficients, what other zero must exist?
Imaginary roots come in...
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Topic 1.5 ยท Fundamental Theorem
How many complex zeros does a degree n polynomial have?
Count multiplicities
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Topic 1.5 ยท Example 1 โ€” Conjugates
What are the conjugates of: 4i, โˆ’i, 2 โˆ’ 3i, โˆ’4 + 2i?
Flip the sign of the imaginary part
6/10
Topic 1.5 ยท Polynomial Inequalities
What does f(x) > 0 mean graphically? What does f(x) < 0 mean?
Think about the x-axis
7/10
Topic 1.5 ยท Sign Chart โ€” 4 Steps
What are the 4 steps for solving a polynomial inequality?
Zeros โ†’ sign chart โ†’ test โ†’ interpret
8/10
Topic 1.5 ยท Example 3
Solve (xโˆ’3)(x+1)(x+4) > 0. Answer in interval notation?
Sign chart with zeros at โˆ’4, โˆ’1, 3
๐Ÿ“„ Page 2 โ€” Answers BACK ยท columns swapped
2/10
โœ“ Even Multiplicity โ€” Bounces
Graph is TANGENT to x-axis (bounces). NO sign change.
Even multiplicity (2,4,6,...) โ†’ graph touches x-axis but does NOT cross. Expression stays same sign on both sides.
1/10
โœ“ Odd Multiplicity โ€” Crosses
Graph PASSES THROUGH x-axis. Sign changes here.
Odd multiplicity (1,3,5,...) โ†’ graph crosses x-axis. Expression changes sign. Mult. 1 = straight cross; mult. 3 = S-curve.
4/10
โœ“ Fundamental Theorem of Algebra
Exactly n complex zeros (counting multiplicities)
A degree n polynomial has exactly n complex zeros counting multiplicities. Some may be imaginary.
3/10
โœ“ Conjugate Pairs
a โˆ’ bi (the complex conjugate)
Imaginary roots come in conjugate pairs. If a+bi is a zero, then aโˆ’bi is automatically also a zero.
6/10
โœ“ f(x) > 0 and f(x) < 0
f(x)>0 โ†’ ABOVE x-axis ยท f(x)<0 โ†’ BELOW x-axis
f(x) = the y-value. Positive y โ†’ above x-axis. Negative y โ†’ below x-axis. f(x) = 0 โ†’ on the x-axis (the zeros).
5/10
โœ“ Example 1 โ€” Conjugates
โˆ’4i ยท i ยท 2+3i ยท โˆ’4โˆ’2i
Flip the sign of the imaginary part only: 4iโ†’โˆ’4i, โˆ’iโ†’i, 2โˆ’3iโ†’2+3i, โˆ’4+2iโ†’โˆ’4โˆ’2i.
8/10
โœ“ Example 3 โ€” Solve > 0
(โˆ’4, โˆ’1) โˆช (3, โˆž)
Zeros: โˆ’4,โˆ’1,3. Signs: โˆ’,+,โˆ’,+. Want positive โ†’ (โˆ’4,โˆ’1) and (3,โˆž). Strict >: open brackets.
7/10
โœ“ 4-Step Sign Chart Method
1) Solve f=0 2) Number line 3) Test values 4) Interpret
1) Find zeros (boundary points). 2) Mark on number line. 3) Test one value per interval โ†’ record +/โˆ’. 4) Select intervals matching the inequality; use [ ] for โ‰ค/โ‰ฅ, ( ) for </>.
๐Ÿ“„ Page 3 โ€” Questions FRONT ยท Sheet 2/2
9/10
Topic 1.5 ยท Successive Differences
How do you find the degree of a polynomial from a table of equal-width input values?
Subtract outputs repeatedly until constant
10/10
Topic 1.5 ยท Even and Odd Functions
How do you test if f is even, odd, or neither? What are the algebraic conditions?
Plug in โˆ’x and compare
๐Ÿ“„ Page 4 โ€” Answers BACK ยท columns swapped
10/10
โœ“ Even and Odd Functions
Even: f(โˆ’x)=f(x) ยท Odd: f(โˆ’x)=โˆ’f(x) ยท Neither: neither
Even: symmetric about y-axis. Odd: symmetric about origin.
Quick test: compare f(โˆ’1) and f(1). If equal โ†’ even. If opposite signs โ†’ odd.
9/10
โœ“ Successive Differences Method
Rounds to reach constant differences = degree
1st differences constant โ†’ degree 1. 2nd constant โ†’ degree 2. 3rd constant โ†’ degree 3.
Requires equal-width input intervals (constant ฮ”x).