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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 ______.
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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.
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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.
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A double root indicates that the root is repeated ______ times.
A double root indicates that the root is repeated ______ times.
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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.
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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?
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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?
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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
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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
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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)
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F ( x ) = − ( x − 2) ( x + 4) /2 , Zeros and ______
F ( x ) = − ( x − 2) ( x + 4) /2 , Zeros and ______
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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 ______
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Factors and their ______
Factors and their ______
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Study Notes
Polynomial Functions - Unit Review
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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
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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
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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
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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
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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
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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
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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
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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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Description
This quiz covers the essential concepts of polynomial functions, including polynomial division, synthetic division, and the Rational Root Theorem. Dive into examples that demonstrate dividing polynomials and evaluating functions, as well as solving polynomial equations. Perfect for students looking to reinforce their understanding of this key mathematical topic.
