Descartes' Rule, Rational Root Theorem, & Finding All Zeros

What information does Descartes' Rule of Signs provide?

• How many solutions to expect

• What kind of solutions to expect (positive, negative, imaginary)

• Possible rational solutions to expect

State the possible number of positive, negative, and imaginary zeros of each function. 

 $f\left(x\right)=27x^6+35x^3+8$  

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State the possible number of positive, negative, and imaginary zeros of each function. 

 $f\left(x\right)=15x^5+25x^4+63x^3+105x^2-60x-100$  

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State the possible number of positive, negative, and imaginary zeros of each function. 

 $f\left(x\right)=5x^7-4x^5-5x^3+4x$  

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What information does the Rational Root Theorem provide?

• How many solutions to expect

• What kind of solutions to expect (positive, negative, imaginary)

• Possible rational solutions

Which of the following is a COMPLETE  list of all possible Rational Zeros?
f(x) = 3x³ + 2x² - 6x + 7

• ±1, ±7, ±1/3, ±7/3
• ±1, 7
• ±1, ±3, ±7, ±7/3
• ±1, ±3, ±7, ±1/3, ±7/3

Tags

CCSS.HSA.APR.B.3

Which of the following is a COMPLETE  list of all possible Rational Zeros?
f(x) = 2x³ + 5x² - 9x + 5

• ±1, ±5, ±1/2, ±5/2
• ±1, ±2, ±5, ±1/2, ±5/2
• ±1, ±5
• ±1, ±2, ±5, ±1/2, ±5/2, ±2/5

Tags

CCSS.HSA.APR.B.2

According to the Rational Root Theorem, what are the all possible rational roots? 
2x³ - 11x² + 12x + 9 = 0

• ±1,±2
• ±1,±3,±9
• ±1,±3±9,±1/2,±3/2,±9/2
• ±1,±2,±1/3,±2/3,±1/9,±2/9

Tags

CCSS.HSA.APR.B.3

Find all zeros without a calculator:

 $5x^6+3x^4-5x^2-3=0$  

•  $\frac{𝑖√15}{5},-\frac{𝑖√15}{5},\ 1,−1,\ 𝑖,\ −𝑖$  

•  $\frac{𝑖√13}{5},-\frac{𝑖√13}{5},\ 1,−1,\ 𝑖,\ −𝑖$  

•  $\frac{𝑖√15}{3},-\frac{𝑖√15}{3},\ 1,−1,\ 𝑖,\ −𝑖$  

Find all zeros. You may use a calculator. 

 $2x^5-10x^4+15x^3-75x^2+28x-140=0$  

•  $3,\ 2i,\ -2i,\ \frac{i\sqrt{13}}{2},\ -\frac{i\sqrt{13}}{2}$  

•  $5,\ 2i,\ -2i,\ \frac{i\sqrt{14}}{2},\ -\frac{i\sqrt{14}}{2}$  

•  $5,\ 3i,\ -3i,\ \frac{i\sqrt{14}}{2},\ -\frac{i\sqrt{14}}{2}$