Solving Polynomial Equations Using Variable Substitution and Factoring
In the following problems, use variable substitution and factoring to find all of the roots of the equation. Let u = x2.
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| 1) Explain how to use variable substitution to find all of the roots of the given polynomial, and then state the roots.
Hint: Let u = x2
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| 2) Solve for x.
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| 3) Solve for x.
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| 4) Solve for x.
Hint: Subtract 14x2 from both sides of the equation. Rearrange the terms on the left side of the equation so they are in decreasing powers of x.
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| 5) Solve for x.
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Rational Root Theorem
Hint: Remember to list the positive and negative rational roots for the function.
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| 6) Use the rational root theorem to list all of the possible rational roots for the function.
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| 7) Use the rational root theorem to list all of the possible rational roots for the function.
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Solving Polynomial Equations Using the Rational Root Theorem
In the following problems, use the Rational Root Theorem to find all of the rational roots of each polynomial equation.
Hint: First use the Rational Root Theorem to find all possible rational roots for the functions. Then use a graphing calculator to determine approximately where the graph crosses the x axis. Then finally test the x values at the points of intersection to see if they are roots of the equation.
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| 8) Solve for x.
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| 9) Solve for x.
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| 10) Solve for x.
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In the following problems, use the rational root theorem and the quadratic equation to solve each polynomial equation.
For a radical answer, express the answer as shown in the example below:
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| 11) Solve for x.
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| 12) Solve for x.
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Proving Polynomial Identities
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| 13) Check the polynomial identity a2 – b2 = (a + b)(a –b) by substituting a = 7 and b = 3.
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Review of Polynomial Functions
In the following problems, evaluate the polynomial expression for the indicated value of “x”.
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In the following problems, use these polynomials to select the simplified form for the given sum or difference. Note: Not all answer choices will be used.
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| 17) Find the sum. Click here to view the choices.
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| 18) Find the difference. Click here to view the choices.
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| 19) Find the sum. Click here to view the choices.
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| 20) Find the difference. Click here to view the choices.
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In the following problems, determine if the given linear expression is a factor of the given polynomial.
Hint: Use substitution to determine whether the given linear expression is a factor of the polynomial.
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| 21) Linear expression a factor of the polynomial?
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| 22) Linear expression a factor of the polynomial?
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In the following problems, divide by using long division and determine if the given solution is true or false. If false, state the correct solution.
Click here to review the unit content explanation for Polynomial Functions.
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| 23) True or False? If false, state the correct solution.
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| 24) True or False? If false, state the correct solution.
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In the following problems, divide using synthetic division and determine if the given solution is true or false. If false, state the correct solution.
Click here to review the unit content explanation for Polynomial Functions.
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| 25) True or False? If false, state the correct solution.
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| 26) True or False? If false, state the correct solution.
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In the following problems, use factoring to find the “real” roots for the given polynomial equation.
Click here to review the unit content explanation for Graphing and Solving Polynomial Equations.
Hint: Factor out the GCF first.
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| 27) Solve for x.
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| 28) Solve for x.
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| 29) Solve for x.
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In the following problems, use a graph, synthetic division, and factoring to find all roots of the equation.
Hint: Factor out the GCF first.
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| 30) Solve for x.
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| 31) Solve for x.
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In the following problems, write a polynomial function, P, in standard form by using the given information.
Click here to review the unit content explanation for Graphing and Solving Polynomial Equations
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| 32) P is of degree 3; P(0) = 2; zeros –1, 1, 2 |
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| 33) P is of degree 4; P(0) = 6; zeros 2, –1, –1, 3 (–1 occurs twice, multiplicity 2) |
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| 34) Extended Learning
Watch the following video, then write a five-sentence paragraph summarizing the video.
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| 35) If you were directed by your school to complete Offline Activities for this course, please enter the information on the Log Entry form. |
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