Finding Real Roots Of Polynomial Equations

FindingRootsof ofFindingRealRealRootsPolynomial EquationsPolynomial EquationsWarm UpLesson PresentationLesson QuizHoltHoltMcDougalAlgebra 2AlgebraAlgebra22HoltMcDougal

Finding Real Roots ofPolynomial EquationsWarm UpFactor completely.1. 2y3 4y2 – 30y 2y(y – 3)(y 5)2. 3x4 – 6x2 – 243(x – 2)(x 2)(x2 2)Solve each equation.3. x2 – 9 0x – 3, 34. x3 3x2 – 4x 0Holt McDougal Algebra 2x –4, 0, 1

Finding Real Roots ofPolynomial EquationsObjectivesIdentify the multiplicity of roots.Use the Rational Root Theorem and theirrational Root Theorem to solvepolynomial equations.Holt McDougal Algebra 2

Finding Real Roots ofPolynomial EquationsVocabularymultiplicityHolt McDougal Algebra 2

Finding Real Roots ofPolynomial EquationsIn Lesson 6-4, you used several methods forfactoring polynomials. As with some quadraticequations, factoring a polynomial equation isone way to find its real roots.Recall the Zero Product Property from Lesson5-3. You can find the roots, or solutions, of thepolynomial equation P(x) 0 by setting eachfactor equal to 0 and solving for x.Holt McDougal Algebra 2

Finding Real Roots ofPolynomial EquationsExample 1A: Using Factoring to Solve PolynomialEquationsSolve the polynomial equation by factoring.4x6 4x5 – 24x4 04x4(x2 x – 6) 0Factor out the GCF, 4x4.4x4(x 3)(x – 2) 0Factor the quadratic.4x4 0 or (x 3) 0 or (x – 2) 0 Set each factorequal to 0.Solve for x.x 0, x –3, x 2The roots are 0, –3, and 2.Holt McDougal Algebra 2

Finding Real Roots ofPolynomial EquationsExample 1A ContinuedCheck Use a graph. Theroots appear to belocated at x 0, x –3,and x 2. Holt McDougal Algebra 2

Finding Real Roots ofPolynomial EquationsExample 1B: Using Factoring to Solve PolynomialEquationsSolve the polynomial equation by factoring.x4 25 26x2x4 – 26 x2 25 0(x2 – 25)(x2 – 1) 0Set the equation equal to 0.Factor the trinomial inquadratic form.(x – 5)(x 5)(x – 1)(x 1) Factor the difference of twosquares.x – 5 0, x 5 0, x – 1 0, or x 1 0x 5, x –5, x 1 or x –1The roots are 5, –5, 1, and –1.Holt McDougal Algebra 2Solve for x.

Finding Real Roots ofPolynomial EquationsCheck It Out! Example 1aSolve the polynomial equation by factoring.2x6 – 10x5 – 12x4 02x4(x2 – 5x – 6) 0Factor out the GCF, 2x4.2x4(x – 6)(x 1) 0Factor the quadratic.2x4 0 or (x – 6) 0 or (x 1) 0 Set each factorequal to 0.x 0, x 6, x –1Solve for x.The roots are 0, 6, and –1.Holt McDougal Algebra 2

Finding Real Roots ofPolynomial EquationsCheck It Out! Example 1bSolve the polynomial equation by factoring.x3 – 2x2 – 25x –50x3 – 2x2 – 25x 50 0Set the equation equal to 0.(x 5)(x – 2)(x – 5) 0Factor.x 5 0, x – 2 0, or x – 5 0x –5, x 2, or x 5Solve for x.The roots are –5, 2, and 5.Holt McDougal Algebra 2

Finding Real Roots ofPolynomial EquationsSometimes a polynomial equation has a factorthat appears more than once. This creates amultiple root. In 3x5 18x4 27x3 0 has twomultiple roots, 0 and –3. For example, the root 0is a factor three times because 3x3 0.The multiplicity of root r is the number of timesthat x – r is a factor of P(x). When a real root haseven multiplicity, the graph of y P(x) touches thex-axis but does not cross it. When a real root hasodd multiplicity greater than 1, the graph “bends”as it crosses the x-axis.Holt McDougal Algebra 2

Finding Real Roots ofPolynomial EquationsYou cannot always determine the multiplicity of aroot from a graph. It is easiest to determinemultiplicity when the polynomial is in factoredform.Holt McDougal Algebra 2

Finding Real Roots ofPolynomial EquationsExample 2A: Identifying MultiplicityIdentify the roots of each equation. State themultiplicity of each root.x3 6x2 12x 8 0x3 6x2 12x 8 (x 2)(x 2)(x 2)x 2 is a factor threetimes. The root –2 hasa multiplicity of 3.Check Use a graph. Acalculator graph showsa bend near (–2, 0). Holt McDougal Algebra 2

Finding Real Roots ofPolynomial EquationsExample 2B: Identifying MultiplicityIdentify the roots of each equation. State themultiplicity of each root.x4 8x3 18x2 – 27 0x4 8x3 18x2 – 27 (x – 1)(x 3)(x 3)(x 3)x – 1 is a factor once, and x 3 is a factor three times.The root 1 has a multiplicityof 1. The root –3 has amultiplicity of 3.Check Use a graph. Acalculator graph showsa bend near (–3, 0) andcrosses at (1, 0). Holt McDougal Algebra 2

Finding Real Roots ofPolynomial EquationsCheck It Out! Example 2aIdentify the roots of each equation. State themultiplicity of each root.x4 – 8x3 24x2 – 32x 16 0x4 – 8x3 24x2 – 32x 16 (x – 2)(x – 2)(x – 2)(x – 2)x – 2 is a factor fourtimes. The root 2 hasa multiplicity of 4.Check Use a graph. Acalculator graph showsa bend near (2, 0). Holt McDougal Algebra 2

Finding Real Roots ofPolynomial EquationsCheck It Out! Example 2bIdentify the roots of each equation. State themultiplicity of each root.2x6 – 22x5 48x4 72x3 02x6 – 22x5 48x4 72x3 2x3(x 1)(x – 6)(x – 6)x is a factor three times, x 1 is a factor once,and x – 6 is a factor two times.The root 0 has a multiplicity of 3. The root –1has a multiplicity of 1. The root 6 has amultiplicity of 2.Holt McDougal Algebra 2

Finding Real Roots ofPolynomial EquationsNot all polynomials are factorable, but the Rational RootTheorem can help you find all possible rational roots ofa polynomial equation.Holt McDougal Algebra 2

Finding Real Roots ofPolynomial EquationsExample 3: Marketing ApplicationThe design of a box specifies that its length is 4inches greater than its width. The height is 1inch less than the width. The volume of the boxis 12 cubic inches. What is the width of the box?Step 1 Write an equation to model the volume ofthe box.Let x represent the width in inches. Then the lengthis x 4, and the height is x – 1.x(x 4)(x – 1) 12 V lwh.x3 3x2 – 4x 12 Multiply the left side.x3 3x2 – 4x – 12 0Holt McDougal Algebra 2Set the equation equal to 0.

Finding Real Roots ofPolynomial EquationsExample 3 ContinuedStep 2 Use the Rational Root Theorem to identify allpossible rational roots.Factors of –12: 1, 2, 3, 4, 6, 12Step 3 Test the possible roots to find one that is actually aroot. The width must be positive, so try only positive rationalroots.Use a synthetic substitution table toorganize your work. The first rowrepresents the coefficients of thepolynomial. The first columnrepresents the divisors and the lastcolumn represents the remainders.Test divisors to identify at least oneroot.Holt McDougal Algebra 2

Finding Real Roots ofPolynomial EquationsExample 3 ContinuedStep 4 Factor the polynomial. The syntheticsubstitution of 2 results in a remainder of 0, so 2is a root and the polynomial in factored form is(x – 2)(x2 5x 6).(x – 2)(x2 5x 6) 0Set the equation equal to 0.(x – 2)(x 2)(x 3) 0Factor x2 5x 6.x 2, x –2, or x –3Set each factor equal to 0,and solve.The width must be positive, so the widthshould be 2 inches.Holt McDougal Algebra 2

Finding Real Roots ofPolynomial EquationsCheck It Out! Example 3A shipping crate must hold 12 cubic feet. Thewidth should be 1 foot less than the length, andthe height should be 4 feet greater than thelength. What should the length of the crate be?Step 1 Write an equation to model the volume ofthe box.Let x represent the length in feet. Then the width isx – 1, and the height is x 4.x(x – 1)(x 4) 12 V lwh.x3 3x2 – 4x 12 Multiply the left side.x3 3x2 – 4x– 12 0Holt McDougal Algebra 2Set the equation equal to 0.

Finding Real Roots ofPolynomial EquationsCheck It Out! Example 3 ContinuedStep 2 Use the Rational Root Theorem to identify allpossible rational roots.Factors of –12: 1, 2, 3, 4, 6, 12Step 3 Test the possible roots to find one that is actually aroot. The width must be positive, so try only positive rationalroots.p1 –4 3 –12Use a synthetic substitution table to qorganize your work. The first row110 4 –12represents the coefficients of thepolynomial. The first column216 50represents the divisors and the last31 14 6 30column represents the remainders.Test divisors to identify at least one41 24 7 84root.Holt McDougal Algebra 2

Finding Real Roots ofPolynomial EquationsCheck It Out! Example 3 ContinuedStep 4 Factor the polynomial. The syntheticsubstitution of 2 results in a remainder of 0, so 2is a root and the polynomial in factored form is(x – 2)(x2 5x 6).(x – 2)(x2 5x 6) 0Set the equation equal to 0.(x – 2)(x 2)(x 3) 0Factor x2 5x 6.x 2, x –2, or x –3Set each factor equal to 0,and solve.The length must be positive, so the lengthshould be 2 feet.Holt McDougal Algebra 2

Finding Real Roots ofPolynomial EquationsPolynomial equations may also have irrational roots.Holt McDougal Algebra 2

Finding Real Roots ofPolynomial EquationsThe Irrational Root Theorem say that irrationalroots come in conjugate pairs. For example, if youknow that 1 is a root of x3 – x2 – 3x – 1 0,then you know that 1 –is also a root.Recall that the real numbers are made up ofthe rational and irrational numbers. You canuse the Rational Root Theorem and theIrrational Root Theorem together to find all ofthe real roots of P(x) 0.Holt McDougal Algebra 2

Finding Real Roots ofPolynomial EquationsExample 4: Identifying All of the Real Roots of aPolynomial EquationIdentify all the real roots of 2x3 – 9x2 2 0.Step 1 Use the Rational Root Theorem to identifypossible rational roots. 1, 2 1, 2, 1 .p 2 and q 2 1, 22Step 2 Graph y 2x3 – 9x2 2 to find the x-intercepts.The x-intercepts are located ator near –0.45, 0.5, and 4.45.The x-intercepts –0.45 and4.45 do not correspond to anyof the possible rational roots.Holt McDougal Algebra 2

Finding Real Roots ofPolynomial EquationsExample 4 Continued1Step 3 Test the possible rational root.21 . The remainder is1Test2 –9 0 22 120, so (x – ) is a factor.1 –4 –222 –8 –4 01The polynomial factors into (x –)(2x2 – 8x – 4).2Step 4 Solve 2x2 – 8x – 4 0 to find theremaining roots.2(x2 – 4x – 2) 0Factor out the GCF, 2Use the quadratic formula to4 16 8 2 6x identify the irrational roots.2Holt McDougal Algebra 2

Finding Real Roots ofPolynomial EquationsExample 4 ContinuedThe fully factored equation is()( 1 2 x – x – 2 6 x – 2 –2 )6 01The roots are, 2 6 , and 2 - 6 .2Holt McDougal Algebra 2

Finding Real Roots ofPolynomial EquationsCheck It Out! Example 4Identify all the real roots of 2x3 – 3x2 –10x – 4 0.Step 1 Use the Rational Root Theorem to identifypossible rational roots. 1, 2, 4 1, 2, 4, 1 . 1, 22 p –4 and q 2Step 2 Graph y 2x3 – 3x2 –10x – 4 to find the xintercepts.The x-intercepts are located ator near –0.5, –1.2, and 3.2.The x-intercepts –1.2 and 3.2do not correspond to any of thepossible rational roots.Holt McDougal Algebra 2

Finding Real Roots ofPolynomial EquationsCheck It Out! Example 4 Continued1Step 3 Test the possible rational root –.211Test–. The remainder2 –3 –10 –4–221 ) is a factor.is0,so(x –1 2 422 –4 –8 01The polynomial factors into (x )(2x2 – 4x – 8).2Step 4 Solve 2x2 – 4x – 8 0 to find theremaining roots.2(x2 – 2x – 4) 0Factor out the GCF, 2Use the quadratic formula to2 4 16x 1 5identify the irrational roots.2Holt McDougal Algebra 2

Finding Real Roots ofPolynomial EquationsCheck It Out! Example 4 ContinuedThe fully factored equation is 1 2 x x –2 ( 1 5 ) x – (1– 5) 1The roots are – , 1 5 , and 1 - 5 .2Holt McDougal Algebra 2

Finding Real Roots ofPolynomial EquationsLesson QuizSolve by factoring.1. x3 9 x2 9x–3, 3, 1Identify the roots of each equation. State themultiplicity of each root.0 and 2 each with2. 5x4 – 20x3 20x2 0multiplicity 23. x3 – 12x2 48x – 64 04 with multiplicity 34. A box is 2 inches longer than its height. The widthis 2 inches less than the height. The volume of thebox is 15 cubic inches. How tall is the box? 3 in.5. Identify all the real roots of x3 5x2 – 3x – 3 0.1, -3 6, -3 - 6Holt McDougal Algebra 2

Polynomial Equations Finding Real Roots of Polynomial Equations Holt Algebra 2 Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 2. Holt McDougal Algebra 2 Finding Real Roots of Polynomial Equations Warm Up Factor comp