Question

Select the correct text in the table. Use the fundamental theorem of algebra to determine whether each statement is sometimes true, always true, or never true.

1. A quadratic function has 2 distinct roots. always sometimes never
2. A cubic function has at least 1 real root. always sometimes never
3. A function with a degree of 5 has 5 roots. always sometimes never

4. A quadratic function can have only 1 complex solution. always sometimes never

Answer 1

MY ANSWER IS CORRECT IN PLATO!!!

1. Sometimes

2. Always

3. Always

4. Never

Explanation:

1. A quadratic function has 2 distinct roots SOMETIMES

2. A cubic Function has at least 1 root ALWAYS

3. A function with a degree of 5 has 5 roots ALWAYS

4. A quadratic function can have only 1 complex solution NEVER

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Answer 2

1. Sometimes

2. Sometimes

3. Always

4. Sometimes

Explanation:

1. Quadratic function : in which maximum power of
`x` is two.

The roots of quadratic function can be either equal or different.

For example:

• 
  `x^(2) -2x+1` will have two equal roots i.e. 1 and 1.
• 
  `x^(2) -3x+2` will have two different roots i.e. 1 and 2.

So, sometimes is the correct answer.

2. Cubic function has atleast 1 real root.

Cubic function has maximum power of
`x` as 3.

If the coefficients are real numbers then atleast 1 real root.

If the coefficients are imaginary in nature, then this is not true.

For example:

Cubic equation
`x^3 +i = 0` does not have any real root.

Cubic equation
`x^3 +1 = 0` has a real root x = -1.

So, it is sometimes true.

3. A function with degree 5 i.e. maximum power of
`x` as 5 will have 5 roots.

It is always true that a function will have number of roots equal to its degree.

4. Quadratic function can have only 1 complex solution.

Two complex solutions are also possible for a quadratic function.

For example:

`x^(2) +1=0` will have two imaginary roots:
`x=i, -i`

It is also possible to have 1 complex solution,

For example:

`(x-1)(x-i) = 0` will have one complex root and one real root.

So, the statement is sometimes true.