Applying properties of operations on complex numbers to solve problems

How many roots or zeros does the equation
f(x) = 5x⁴-8x³+4x²-6x+3 have?

• 3
• 4
• 5
• 0

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CCSS.HSN.CN.C.9

⅖(x + 3)² + 8 = -32

• x = 3 ± 10i

• x = ±13i

• x = -3 ± 10i

• x = ±i√10

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CCSS.HSN.CN.C.9

What is the least degree of the following polynomial that has 4 real roots and 2 imaginary roots?

• 4

• 2

• 8

• 6

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CCSS.HSN.CN.C.9

Write the polynomial in standard form given the following zeros. x = -2, 1, 4

• f(x) = (x+2)(x-1)(x-4)

• f(x) =x³ - 3x² - 6x + 8

• f(x) = x³ + 8

• f(x) = x³ - 3x² + 8

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CCSS.HSN.CN.C.9

Can a 7th degree polynomial have 9 real roots?

• Yes, but only if some roots are complex.

• Yes, if all roots are distinct.

• No, it must have exactly 7 complex roots.

• No, it can have at most 7 real roots.

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CCSS.HSN.CN.C.9

• 1

• 3

• 7

• 2

Answer explanation

The Fundamental Theorem of Algebra states that a polynomial of degree n has a total of n real and complex solutions.

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CCSS.HSN.CN.C.9

Describe the Irrational Root Theorem: (If a polynomial has a root of

 $a+\sqrt{b}$  , what other root must the equation have?)

•  $a-\sqrt{b}$  

•  $a-b$  

•  $-b$  

•  $a+\sqrt{b}$  

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CCSS.HSN.CN.C.9

If a quadratic equation can be factored and each factor contains only real numbers, then it cannot have an imaginary solution.

• True

• False

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CCSS.HSN.CN.C.9

The largest exponent determines the degree of the polynomial.

• True
• False

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CCSS.HSN.CN.C.9

This is a degree 4 function. Describe the number and types of roots.

• 2 real roots (no repeats)

• 2 real roots ( 1 unique with multiplicity 2), 2 complex roots

• 2 complex roots (no repeats)

• 2 real roots (1 unique with multiplicity 2)

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CCSS.HSN.CN.C.9