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Questions and Answers
To divide polynomials using long division, you must first divide the ______ term of the dividend by the leading term of the divisor.
To divide polynomials using long division, you must first divide the ______ term of the dividend by the leading term of the divisor.
leading
Synthetic division is a shortcut method for dividing a polynomial by a ______ linear expression.
Synthetic division is a shortcut method for dividing a polynomial by a ______ linear expression.
linear
When using synthetic division, the ______ of the divisor is used as the test value.
When using synthetic division, the ______ of the divisor is used as the test value.
opposite
The Remainder Theorem states that if a polynomial, P(x), is divided by x - a, then the remainder is equal to ______.
The Remainder Theorem states that if a polynomial, P(x), is divided by x - a, then the remainder is equal to ______.
The Factor Theorem states that a binomial (x - a) is a ______ of a polynomial P(x) if and only if P(a) = 0.
The Factor Theorem states that a binomial (x - a) is a ______ of a polynomial P(x) if and only if P(a) = 0.
To find the possible rational roots of a polynomial equation, we use the ______ Theorem.
To find the possible rational roots of a polynomial equation, we use the ______ Theorem.
A double root indicates that the root is repeated ______ times.
A double root indicates that the root is repeated ______ times.
The volume of a rectangular prism can be calculated by multiplying the ______, width, and height.
The volume of a rectangular prism can be calculated by multiplying the ______, width, and height.
The profit, P (in millions of dollars) for a manufacturer of Tacky Thingamabobs can be modeled by the function, P ( x ) =− x 4 + 7 x3 − 12 x 2 − 4 x + 38 where x is the number of Tacky Thingamabobs produced (in millions). Currently, the company produces 4 million widgets and makes a profit of $22,000,000. What ______ number of widgets could the company produce and still make the same profit?
The profit, P (in millions of dollars) for a manufacturer of Tacky Thingamabobs can be modeled by the function, P ( x ) =− x 4 + 7 x3 − 12 x 2 − 4 x + 38 where x is the number of Tacky Thingamabobs produced (in millions). Currently, the company produces 4 million widgets and makes a profit of $22,000,000. What ______ number of widgets could the company produce and still make the same profit?
The depth of the pool is x meters, the width of the pool is 8 more than twice the depth and the length is five times the depth. If the volume of the pool is 3600 m3, then what is the ______ of the pool?
The depth of the pool is x meters, the width of the pool is 8 more than twice the depth and the length is five times the depth. If the volume of the pool is 3600 m3, then what is the ______ of the pool?
Find the polynomial function of least degree in ______ form given zeros of 3 and − 1 − 3i
Find the polynomial function of least degree in ______ form given zeros of 3 and − 1 − 3i
Find the polynomial function of least degree in ______ form given zeros of -5 and − 1 + 2 3
Find the polynomial function of least degree in ______ form given zeros of -5 and − 1 + 2 3
Write the function that has the given the points in ______ form.( −2, 0 ) , (−5, 0), (1, 0), (0, 0), (−4, −5)
Write the function that has the given the points in ______ form.( −2, 0 ) , (−5, 0), (1, 0), (0, 0), (−4, −5)
F ( x ) = − ( x − 2) ( x + 4) /2 , Zeros and ______
F ( x ) = − ( x − 2) ( x + 4) /2 , Zeros and ______
F ( x ) = x 6 − x5 − 2 x 4 + 2 x3 + x 2 − x, End ______
F ( x ) = x 6 − x5 − 2 x 4 + 2 x3 + x 2 − x, End ______
Factors and their ______
Factors and their ______
Flashcards
Volume of a Pool
Volume of a Pool
The volume can be calculated as depth × width × length.
Profit Function
Profit Function
P(x) = −x^4 + 7x^3 − 12x^2 − 4x + 38 models the profit.
Zeros of Polynomial
Zeros of Polynomial
Zeros are values of x that make f(x) = 0.
End Behavior
End Behavior
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Extrema
Extrema
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Multiplicity
Multiplicity
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Factored Form
Factored Form
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Polynomial Degree
Polynomial Degree
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Polynomial Division
Polynomial Division
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Synthetic Division
Synthetic Division
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Rational Roots Theorem
Rational Roots Theorem
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Factoring a Polynomial
Factoring a Polynomial
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Evaluating a Polynomial
Evaluating a Polynomial
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Polynomial Zeros
Polynomial Zeros
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Degree of a Polynomial
Degree of a Polynomial
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Evaluation of P(x)
Evaluation of P(x)
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Study Notes
Polynomial Functions - Unit Review
- Section 1: Polynomial Division
- Long division is used to divide polynomials
- Example 1: (3x² - 2x² + 3x - 1) ÷ (x² + 3)
- Example 2: (2x³ - 3x² + 2x² + 3x - 2) ÷ (x² - 2x + 1)
Section 2: Applications of Synthetic Division
- Synthetic Division: A method for dividing polynomials, especially when the divisor is in the form (x - c).
- Example 3: (x² - 2x + 5) ÷ (x - 3)
- Example 4: (x³ - 3x² + 5x - 1) ÷ (x + 2)
- Evaluating Functions using Synthetic Division:
- Example 5: Evaluate P(x) = x³ – 3x² - 2x + 5 for P(2)
- Example 6: Evaluate P(x) = x³ – 3x² - 2x + 5 for P(-1)
Section II: Solving Polynomial Equations
-
Rational Root Theorem: Used to find possible rational roots of polynomial equations.
- Example 9: 2x³ - 3x² + 2x = 8
- Example 10: 4x³ - 2x² + 3x - 10 = 0
-
Determining Roots/Zeros:
- Given a factor, find the other roots.
- Example 11: 6x³ - 11x² - 3x + 2 = 0 ; (x - 2) is a factor
- Example 12: x⁴ + 3x³ + x² - 3x² - 2x = 0 ; x = -1 is a double root
Section III: Practical Problems
- Word Problems involving Polynomials
- Example 17: A pool’s volume is given by a polynomial function; how deep is the pool?
- Example 18: A profit function relates profit to the number of items; at what production level does profit equal $22,000,000?
Section IV: Graphing Polynomials
- Graphing Polynomials:
- Example 19: Sketch f(x) = -(x - 2)²(x + 4)
- Analyze zeros, multiplicity, end-behavior, and extrema
- Example 20: Sketch f(x) = x⁴ - x³ - 2x² + 2x² + x² - x
- Analyze zeros, multiplicity, end-behavior, and extrema
Section V: Writing Rules for Polynomial Functions
- Polynomial Function Rules:
- Find a polynomial function from given zeros.
- Example 21: Find the polynomial function of least degree given zeros of 3 and −1 − 3i
- Example 22: Find a polynomial function given zeros of -5 and −1 + 2\√3
- Example 23: Write the equation from the graph
- Example 24: Write the equation given the point values
Section VI: Interpreting Graphs
- Graph Interpretation:
- Example 25 and 26: Analyze the graphs for zeros, multiplicity, end-behavior, leading coefficients, and degree.
- Analyze features of the graph
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