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Chapter 5

Polynomials

Polynomial functions have graphs that are smooth curves. They go from negative infinity to positive infinity in a nice, flowing fashion with no abrupt changes of direction. Pieces of polynomial functions are helpful when modeling physical situations, such as the height of a rocket shot in the air or the time a person takes to swim a lap depending on their age.

Most of the focus on polynomial functions is in determining when the function changes from negative values to positive values or vice versa. Also of interest is when the curve hits a relatively high point or relatively low point. Some good algebra techniques go a long way toward studying these characteristics of polynomial functions.

The Problems You’ll Work On

In this chapter, you’ll work with polynomial functions in the following ways:

 Solving quadratic equations by factoring or using the quadratic formula

 Rewriting quadratic equations by completing the square

 Factoring polynomials by using grouping

 Looking for rational roots of polynomials by using the rational root theorem

 Counting real roots with Descartes’s rule of signs

 Using synthetic division to quickly compute factors

 Writing equations of polynomials given roots and other information

 Graphing polynomials by using end-behavior and the factored form

What to Watch Out For

Don’t let common mistakes trip you up; keep in mind that when working with polynomial functions, your challenges will include

 Watching the order of operations when using the quadratic formula

 Adding to both sides when completing the square

 Remembering to insert zeros for missing terms when using synthetic division

 Recognizing the effect of imaginary roots on the graph of a polynomial

Using Factoring to Solve Quadratic Equations

251–260 Find the solution set for the equation by factoring.

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Solving Quadratic Equations by Using the Quadratic Formula

261–265 Use the quadratic formula to solve the equation.

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Using Completing the Square to Solve Quadratic Equations

266–270 Solve the equation by completing the square.

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Solving Polynomial Equations for Intercepts

271–276 Find the real roots of the polynomial.

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Using Factoring by Grouping to Solve Polynomial Equations

277–280 Find the real roots of the polynomial by using factoring by grouping.

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Applying Descartes’s Rule of Signs

281–285 Use Descartes’s rule of signs to determine the possible number of positive and negative real roots of the function.

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Listing Possible Roots of a Polynomial Equation

286–290 Use the rational root theorem to list the possible rational roots of the function.

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Dividing Polynomials

291–295 Perform the division by using long division. Write the remainder as a fraction.

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Using Synthetic Division to Divide Polynomials

296–298 Use synthetic division to find the quotient and remainder.

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Checking for Roots of a Polynomial by Using Synthetic Division

299–300 Use synthetic division to determine which of the given values is a zero of the function.

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Possible zeros: –3, 3, –6, 6, 1

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Possible zeros: –8, –2, –1, 4, 8

Writing Polynomial Expressions from Given Roots

301–305 Find the lowest order polynomial with 1 as the leading coefficient with the listed values as its zeros.

301. –3, 2, 3

302. –4, –3, 0, 1, 2

303. , , –1, 2

304. , , –1, and –4

305. ,

Writing Polynomial Expressions When Given Roots and a Point

306–310 Find the lowest-order polynomial that has the listed values as its zeros and whose graph passes through the given point.

306. Zeros: –3, 1, 2

Point:

307. Zeros: –4, –2, 3

Point:

308. Zeros: –3, –2, 1, 2

Point:

309. Zeros: ,

Point:

310. Zeros: 2 + i, 2 – i, –3, 2

Point:

Graphing Polynomials

311–316 Graph the polynomial by determining the end behavior, finding the x- and y-intercepts, and using test points between the x-intercepts.

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Writing Equations from Graphs of Polynomials

317−320 Write an equation for the given polynomial graph.

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Illustration by Thomson Digital

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Illustration by Thomson Digital

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Illustration by Thomson Digital

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Illustration by Thomson Digital