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Questions and Answers
What is the leading term of the polynomial $6x^3 + 2x^2 - 5$?
What is the leading term of the polynomial $6x^3 + 2x^2 - 5$?
- $-5$
- $2x^3$
- $6x^3$ (correct)
- $2x^2$
What is the degree of the polynomial $4x^5 - 3x^2 + 7x - 8$?
What is the degree of the polynomial $4x^5 - 3x^2 + 7x - 8$?
- 3
- 5 (correct)
- 2
- 7
Which statement is true regarding synthetic division?
Which statement is true regarding synthetic division?
- It is used when the divisor is a polynomial of degree 2.
- It always produces a remainder of zero.
- It simplifies the division process for polynomials. (correct)
- It can only be used with numerical coefficients.
According to the Remainder Theorem, what is the remainder when dividing a polynomial $P(x)$ by $(x - c)$?
According to the Remainder Theorem, what is the remainder when dividing a polynomial $P(x)$ by $(x - c)$?
What does it indicate if a polynomial has an odd degree with a positive leading coefficient?
What does it indicate if a polynomial has an odd degree with a positive leading coefficient?
Which of the following describes the constant term in the polynomial $3x^4 + 0x^3 + 7x - 5$?
Which of the following describes the constant term in the polynomial $3x^4 + 0x^3 + 7x - 5$?
What is the factor theorem used for?
What is the factor theorem used for?
In polynomial division using long division, what does 'B' stand for in the DMSBR algorithm?
In polynomial division using long division, what does 'B' stand for in the DMSBR algorithm?
Flashcards
Leading Term
Leading Term
The term in a polynomial with the highest exponent.
Leading Coefficient
Leading Coefficient
The number in front of the leading term.
Degree of a Polynomial
Degree of a Polynomial
The highest exponent in a polynomial.
Remainder Theorem
Remainder Theorem
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Factor Theorem
Factor Theorem
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Odd Degree Polynomial
Odd Degree Polynomial
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Even Degree Polynomial
Even Degree Polynomial
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End Behavior
End Behavior
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Study Notes
Polynomials
- Polynomials are algebraic expressions using variables with whole number exponents and coefficients.
Key Concepts
- Leading Term: The term with the highest exponent.
- Leading Coefficient: The coefficient of the leading term.
- Degree: The highest exponent of the variable.
- Constant Term: The term without a variable.
Equations/Functions
- An equation sets two expressions equal.
- A function relates input to output.
Polynomial Division
- Long Division (DMSBR): Divide, Multiply, Subtract, Bring Down, Repeat.
- Synthetic Division (DMAD): Divide, Multiply, Add, Drop, and Repeat.
Theorems
- Remainder Theorem: When dividing a polynomial by , the remainder is.
- Factor Theorem: If then is a factor of.
- Rational Root Theorem: Possible roots of are , where is a factor of the constant term and is a factor of the leading coefficient.
End Behavior
- Odd Degree:
- Positive Leading Coefficient: Rises right, falls left
- Negative Leading Coefficient: Falls right, rises left
- Even Degree:
- Positive Leading Coefficient: Rises on both ends.
- Negative Leading Coefficient: Falls on both ends.
Graphing Polynomials
- Use the leading term to determine the general shape.
- Use the degree to identify possible turning points.
- Use roots and their multiplicities to find where the graph crosses or touches the x-axis.
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