Podcast
Questions and Answers
What is the result of synthetic division when dividing a polynomial by (x - 2)?
What is the result of synthetic division when dividing a polynomial by (x - 2)?
- A new polynomial without remainder
- Cannot be determined
- A remainder only
- A quotient with a remainder (correct)
Which of the following describes the Rational Zero Theorem?
Which of the following describes the Rational Zero Theorem?
- A technique for approximating polynomial roots
- A procedure for finding derivatives
- A method to find irrational roots
- A way to test possible rational zeros (correct)
Which transformations would result in a reflection of a graph across the x-axis?
Which transformations would result in a reflection of a graph across the x-axis?
- Changing the sign of the y-values
- Reducing the degree of the polynomial
- Multiplying the function by -1 (correct)
- Adding a constant to the x-values
What is one method to find x-intercepts of a polynomial function?
What is one method to find x-intercepts of a polynomial function?
Which of the following properly expresses the product property of exponents?
Which of the following properly expresses the product property of exponents?
How do you find vertical asymptotes of a rational function?
How do you find vertical asymptotes of a rational function?
In the context of factoring polynomials, what does finding the roots of a polynomial mean?
In the context of factoring polynomials, what does finding the roots of a polynomial mean?
Which statement about the leading coefficient of a polynomial is true concerning transformations?
Which statement about the leading coefficient of a polynomial is true concerning transformations?
Flashcards
Synthetic Division
Synthetic Division
A method for dividing a polynomial by a linear expression by using only the coefficients of the polynomials, simplifying the division process, and making it faster and easier.
Rational Zero Theorem
Rational Zero Theorem
A theorem that provides a list of all possible rational roots of a polynomial equation. It helps to narrow down the search for real number solutions.
Graph Transformations
Graph Transformations
Analyzing how changes to the equation of a function (like adding a constant, multiplying by a factor, or negating the function) affect the graph of the function. These changes can involve shifts, stretches, compressions, and reflections.
Factoring Polynomials
Factoring Polynomials
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Exponent Properties
Exponent Properties
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Radical Properties
Radical Properties
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Intercepts of a Function
Intercepts of a Function
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Asymptotes of a function
Asymptotes of a function
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Study Notes
Synthetic Division
- Used to divide polynomials by a linear factor of the form (x - c)
- Efficient alternative to polynomial long division
- Necessary for finding roots and factors of a polynomial
- Remainder theorem – remainder when dividing P(x) by (x - c) is equal to P(c).
Rational Root Theorem
- Used to find possible rational roots of a polynomial equation.
- Factors of the constant term divided by factors of the leading coefficient
- Focuses on the possible rational values of x that could be roots.
Binomial Theorem
- Used to expand binomial expressions
- Formula for (a + b)ⁿ
- Terms are determined by the binomial coefficients, combination(n, k).
Graphing Transformations
- Vertical shifts involve adding or subtracting a constant to the function
- Horizontal shifts change the input value
- Stretches/compressions alter the function's vertical or horizontal scaling.
- Reflections change the orientation of the function.
- Recognizing parent functions is key to recognizing transformations.
Polynomial Operations
- Product of powers: multiply expressions with the same base by adding the exponents
- Quotient of powers: divide expressions with the same base by subtracting the exponents
- Power of a power: raise an exponential expression to a power by multiplying the exponents
- Zero exponent: any non-zero number raised to the power of zero equals one
- Negative exponent: a number raised to a negative exponent is equal to the reciprocal of the base raised to the positive form of the exponent.
Polynomials
- Degree of a polynomial is the highest power of the variable
- Leading coefficient is the coefficient of the highest degree term
- End behavior describes the graph's behavior as x approaches infinity and negative infinity
- Graphs of polynomials can intersect or touch x-axis. Number of intersections or tangents = to degree in polynomial.
Roots/Zeros of a Polynomial
- Values of x for which P(x) = 0
- Rational roots are possible rational values as given by Rational Root Theorem
- Synthetic division can help find out if a given number is a root.
Asymptotes
- Vertical asymptotes occur where the denominator of a rational function is zero, and numerator is not zero
- Horizontal asymptotes indicate the long-run behavior of the graph
- Slant (or oblique) asymptotes occur when the degree of the numerator is one more than the degree of the denominator. It is found by performing polynomial division.
Various types of functions
- Quadratic: 𝑦=𝒂𝒙² + 𝒃𝒙 + 𝒄
- Cubic: 𝑦=𝒂𝒙³ + 𝒃𝒙² + 𝒄𝒙 + 𝒅
- Square root: 𝑦=√𝒙
- Cube root: 𝑦=³√𝒙
Additional Concepts
- Remainder Theorem: Remainder when dividing a polynomial by (x – c) is equal to P(c)
- Factor Theorem: (x – c) is a factor of P(x) if and only if P(c) = 0.
- Possible rational roots(m/n): Possible values that satisfy polynomial. (factors of constant term) / (factors of leading coefficient).
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