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Questions and Answers
For the polynomial $y= (x+2)(x-1)(x-3)$, one of the zeros is 2.
For the polynomial $y= (x+2)(x-1)(x-3)$, one of the zeros is 2.
False (B)
If a polynomial has zeros 0, 5, and 10, then one of its linear factors must be $x - 5$.
If a polynomial has zeros 0, 5, and 10, then one of its linear factors must be $x - 5$.
True (A)
The expression $a - 5$ is a trinomial.
The expression $a - 5$ is a trinomial.
False (B)
The expression $x^3 + 3x^2 - 6$ has a degree of 3.
The expression $x^3 + 3x^2 - 6$ has a degree of 3.
The expression $5x^2 - 16 + 8x^3$ has as its leading term $5x^2$.
The expression $5x^2 - 16 + 8x^3$ has as its leading term $5x^2$.
This polynomial is in standard form: $1 + 4x + 8x^2$.
This polynomial is in standard form: $1 + 4x + 8x^2$.
Where a graph changes direction is called a direction-changing point.
Where a graph changes direction is called a direction-changing point.
The direction of a graph to the far left and to the far right is called end behavior.
The direction of a graph to the far left and to the far right is called end behavior.
The end behavior of a polynomial function can be determined by its leading term.
The end behavior of a polynomial function can be determined by its leading term.
The end behavior of $f(x) = 7x^3 - 5x + 13$ is down and up.
The end behavior of $f(x) = 7x^3 - 5x + 13$ is down and up.
The number 2744 is a perfect cube.
The number 2744 is a perfect cube.
If you calculate the discriminant of a polynomial equation to be a negative number, then you will have at least one imaginary solution.
If you calculate the discriminant of a polynomial equation to be a negative number, then you will have at least one imaginary solution.
It does not make sense to describe the end behavior of a polynomial function as up, down, and up.
It does not make sense to describe the end behavior of a polynomial function as up, down, and up.
If a turning point of the graph of a polynomial function is a relative maximum, that point will look like it is at the top of a hill, not at the bottom of a valley.
If a turning point of the graph of a polynomial function is a relative maximum, that point will look like it is at the top of a hill, not at the bottom of a valley.
If you find the solutions of a polynomial equation and one solution is $4 + 3i$, then another solution will be $4 - 3i$.
If you find the solutions of a polynomial equation and one solution is $4 + 3i$, then another solution will be $4 - 3i$.
To say that a polynomial function has as one of its zeros -2 of multiplicity 3 means that the function has $x + 2$ as a factor three times: $(x+2)(x+2)(x+2)$.
To say that a polynomial function has as one of its zeros -2 of multiplicity 3 means that the function has $x + 2$ as a factor three times: $(x+2)(x+2)(x+2)$.
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Study Notes
Polynomial Zeros and Factors
- For the polynomial ( y = (x+2)(x-1)(x-3) ), 2 is not one of its zeros.
- If a polynomial has zeros at 0, 5, and 10, then ( x-5 ) must be one of its linear factors.
Types of Polynomial Expressions
- The expression ( a-5 ) is not a trinomial; it is a binomial.
- The expression ( x^3 + 3x^2 - 6 ) is classified with a degree of 3, indicating its highest exponent.
Leading Terms and Standard Form
- The leading term of the expression ( 5x^2 - 16 + 8x^3 ) is not ( 5x^2 ), but ( 8x^3 ), as it has the highest degree.
- A polynomial is in standard form when its terms are arranged in descending order of degree. The expression ( 1 + 4x + 8x^2 ) is not in standard form.
Graphing Polynomials
- A point where a graph changes direction is misnamed; the correct term is "turning point."
- The overall trend of a graph towards the far left and far right is referred to as its "end behavior."
End Behavior and Leading Terms
- The end behavior of a polynomial function is determined by its leading term, which affects how the graph behaves as ( x ) approaches positive or negative infinity.
- For ( f(x) = 7x^3 - 5x + 13 ), the end behavior is characterized as down on the left side and up on the right side.
Perfect Cubes and Discriminants
- The number ( 2744 ) qualifies as a perfect cube.
- When using the quadratic formula, a negative discriminant indicates at least one imaginary solution.
Describing End Behavior
- It is illogical to describe the end behavior of a polynomial function as "up, down, and up"; instead, the description would typically indicate consistent trends.
- A relative maximum turning point appears visually as the peak of a hill, while a minimum turning point resembles the bottom of a valley.
Complex Solutions in Polynomials
- If a polynomial equation has a solution in the form ( 4 + 3i ), it necessarily also has ( 4 - 3i ) as another solution due to complex conjugate pairs.
- Describing a polynomial function with a zero of -2 of multiplicity 3 implies ( (x + 2) ) is a factor three times: ( (x + 2)(x + 2)(x + 2) ).
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