Ready To Go On? Skills Intervention 6-1 Polynomials
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1 6A Read To Go On? Skills Intervention 6- Polnomials Find these vocabular words in Lesson 6- and the Multilingual Glossar. Vocabular monomial polnomial degree of a monomial degree of a polnomial leading coefficient binomial trinomial polnomial function Classifing Polnomials Rewrite the polnomial in standard form. Then identif the leading coefficient, degree, and number of terms. Name the polnomial. A polnomial is written in standard form when its terms are written in order b degree. leading coefficient Write the given polnomial in standard form. degree of polnomial Identif the leading coefficient. Identif the degree of the leading coefficient. A term is separated b addition or subtraction signs. How man terms are there? A polnomial is named b its largest degree. Is the polnomial quadratic or cubic? Adding and Subtracting Polnomials Add or subtract. Write our answer in standard form. A. ( ) ( ) To add verticall, write each polnomial in form and align terms First polnomial Second polnomial Add. B. ( ) ( ) To subtract horizontall, ou add the. ( ) ( 3 2 5) Change the sign and distribute the negative. (0 3 3 ) ( 2 2 ) Rearrange so like terms are together Add. 9 Holt Algebra 2
2 6A Read To Go On? Skills Intervention 6-2 Multipling Polnomials Multipling Polnomials Find each product. To multipl an two polnomials, use the Propert to multipl each term in the second polnomial b each term in the polnomial. A. Multipl horizontall. (2 )( ) Distribute 2, and then. 2( 3 ) (2 2 ) (6) 2() ( 3 ) (2 2 ) () When ou multipl, ou eponents. 4 ( 3 3 ) (2 2 2 ) ( 6) Combine like terms Simplif. B. Multipl verticall. ( 2 6 5)( 4) Multipl b Multipl b. 3 2 Combine like terms. Using Pascal s Triangle to Epand Binomial Epressions Epand the epression (3 4 ) 3. Each row of Pascal s triangle gives the binomial epansion. of the corresponding Using Pascal s Triangle, what are the coefficients for n 3 (row 4)? The eponents on the first term and the eponents on the second term decrease. What is the first term in the epression (3 4 ) 3? second term? Complete the epansion. [ (3 ) 3 ( 4) 0 ] [ 3(3 ) 2 ( 4) ] [ (3) ( 4) ] [ (3 ) ( 4) ] [ 3 ()] [ 3 2 ( 4)] [ ( )] [ () ] Simplif Simplif. Pascal s Triangle Holt Algebra 2
3 6A Read To Go On? Skills Intervention 6-3 Dividing Polnomials Find this vocabular word in Lesson 6-3 and the Multilingual Glossar. Using Snthetic Division to Divide b a Linear Binomial Divide. ( ) ( 2) What are the coefficients of the dividend?,,, The divisor is written in the form a. What is the value of a? Write the coefficients of the dividend and the value for a in the upper left corner Vocabular snthetic division Bring down the first coefficient () and write it below the horizontal bar. Multipl 2 b to get 2. Write the product under the net coefficient and add. Repeat the steps (multipl, write the product under the net coefficient and add) with the remaining numbers. The answer is 2. Using Snthetic Substitution Use snthetic substitution to evaluate P () for 2. You can use snthetic division to evaluate polnomials. The process is eactl the same as snthetic division, but the final answer is interpreted differentl. If ou are missing terms what do ou do? What are the coefficients of the polnomial?,, 2,, Write the coefficients Bring down the first coefficient () and write it below the horizontal bar. Complete the snthetic division b multipling and adding. Check: Substitute 2 for in P() The answer should be our remainder. P( ) Does our answer check? 93 Holt Algebra 2
4 6A Read To Go On? Skills Intervention 6-4 Factoring Polnomials Factoring b Grouping Factor Group the terms. The common monomial for the first group is 2, what is the common monomial for the second group? 2 Factor common monomials from each group. What is the common binomial factor? 2 Factor out the common binomial. Factoring the Sum or Difference of Two Cubes Factor each epression. What is the general form for the sum of two cubes? a 3 b 3 (a )(a 2 b 2 ) What is the general form for the difference of two cubes? a 3 b 3 (a )(a 2 b 2 ) A Do ou have a sum or difference of two cubes? How can ou write 25 3 as a cube? 3 How can ou write 3 as a cube? Rewrite as a sum of cubes. 3 What represents a? What represents b? Use the rule. a 3 b Simplif. 2 B Do ou have a sum or difference of two cubes? How can ou write 6 as a cube? What represents a? What represents b? Rewrite as a difference of cubes How can ou write 729 as a cube? Use the rule. a 3 b Simplif Factor 2 9 as a difference of two squares Holt Algebra 2
5 6A Read To Go On? Quiz 6- Polnomials Rewrite each polnomial in standard form. Then identif the leading coefficient, degree, and number of terms. Name the polnomial Leading coefficient Leading coefficient Degree Number of terms Degree Number of terms Name Name Leading coefficient Leading coefficient Degree Number of terms Degree Number of terms Name Name Add or subtract. Write our answer in standard form. 5. (4 2 3) (5 2 4) 6. ( ) ( ) 7. ( ) ( ) 8. The profit on -units of a product can be modeled b P() Evaluate P () for 50, and describe what the value represents. Graph each polnomial function on a calculator. Describe the graph, and identif the number of real zeros. 9. g ( ) h ( ) Multipling Polnomials Find each product.. 3(2 2 5) 2. (a 2b)(4ab b 2 ) 95 Holt Algebra 2
6 6A Read To Go On? Quiz continued (3 2)( ) Epand each epression. 5. ( 4 ) 4 6. ( 3 ) 3 7. (5 2 ) 4 8. Find the polnomial epression in terms of for the volume of the rectangular prism shown Dividing Polnomials Divide. 9. ( ) (5 ) 20. ( ) ( 2) 4 Use snthetic substitution to evaluate the polnomial for the given value. 2. P() for P() for Factoring Polnomials Factor each epression The volume of a bo is modeled b the function V() Identif the values of for which the volume is 0 and use the graph to factor V() Holt Algebra 2
7 6A Read To Go On? Enrichment Packaging A package for a new to is being designed out of a piece of heav cardboard that is 42 square inches. The base has a width of 6 inches and the length is 6 inches more than the height. The piece of cardboard also has tabs that are inch wide on each side of the ends and a -inch wide tab on the lid. Top Bottom Find the height,, and the volume of the bo b following the steps.. Label all remaining dimensions in the figure. Top 2. Find an epression for each of the rectangular regions. Bottom 3. Write an epression for the surface area b adding together each of the rectangular areas. SA 4. Simplif the epression for surface area. 5. Write an equation for the surface area. 6. Solve the equation for. 7. Find the approimate volume of the package. 97 Holt Algebra 2
8 Read To Go On? Skills Intervention 6-5 Finding Real Roots of Polnomial Equations Find this vocabular word in Lesson 6-5 and the Multilingual Glossar. Using Factoring to Solve Polnomial Equations Solve the polnomial equation, , b factoring What is the greatest common factor? 3 2 Factor out the GCF. 3 Factor the quadratic. 0, 2, or Set each factor equal to 0. 0 Solve for. The roots are and. Vocabular multiplicit Identifing Multiplicit Identif the roots of each equation. State the multiplicit of each root. The multiplicit of a root is the number of times that r is a factor of P(). When a real root has an does not cross it. When a real root has an crosses the -ais. multiplicit, the graph touches the -ais but multiplicit greater than, the graph bends as it A Factor out the greatest common factor. 0 Factor the quadratic. 2 0, 0, or 0 Set each factor equal to 0. or Solve for. is a factor times. The root 0 has multiplicit of. 7 is a factor times. The root 7 has multiplicit of. Check: Graph the function on a graphing calculator. Does the graph show that touches or bends at the -ais? B Factor out the GCF. ( 3)( 4)( 4) Factor. 3 is used as a factor time. The root 3 has a multiplicit of. 4 is used as a factor times. The root has a multiplicit of. 98 Holt Algebra 2
9 Read To Go On? Problem Solving Intervention 6-5 Finding Real Roots of Polnomial Equations In a previous lesson ou used several was to factor polnomials. As with some quadratic equations, factoring a polnomial equation is one wa to find its real roots. The earl profit of a compan in thousands of dollars can be modeled b P(t ) t 4 74 t 2 225, where t is the number of ears since 200. Factor the polnomial to find the ears in which the profit was 0. Understand the Problem. What needs to be done in order to find the ear in which the profit is 0? 2. What is the given equation? Make a Plan 3. How man roots will the equation have? 4. Write the equation when the profit is Is there a greatest common factor? 6. Is the polnomial a sum or difference of cubes? 7. Is the polnomial a difference of two squares? 8. What are the factors of 225 that when added equal 74? and Solve 9. Factor the polnomial. ( t 2 )( t 2 ) 0 (t )(t )(t )(t ) 0 t 7 0, 0, t 5 0, 0 Use the Zero Product Propert. t, t, t, t Solve for t. 0. What are the roots of the polnomial?,,,. Can the negative roots be used? Wh? 2. In what ears will the profit be 0? and 200 Look Back 3. Graph P(t ) t 4 74 t on a graphing calculator. 4. Locate where the graph crosses the -ais. Does it cross the -ais at 5 and 7? 99 Holt Algebra 2
10 Read To Go On? Skills Intervention 6-6 Fundamental Theorem of Algebra Writing Polnomial Functions Given Zeros Write the simplest polnomial function with roots 3, 3, and 4. If r is a root of P(), then r is a factor of P(). P() ( 3)( )( ) Write each root as a factor. ( 2 9)( ) Multipl the first two binomials. ( 2 )( ) Simplif Multipl the two binomials. Finding All Roots of a Polnomial Equation Solve b finding all roots. The polnomial is of degree, so there are roots for the equation. Step Use the Rational Root Theorem to identif all possible rational roots. p q,,, 4, 6, p and q Step 2 Graph to find the The real roots appear to be at or near and. Step 3 Test the possible real roots. roots The polnomial factors into ( )( 3 2 2). Test the other root in the cubic polnomial The polnomial factors into ( )( 3)( ) 0. 2 Step 4 Solve 2 0 to find the remaining roots The solutions are,,, and. Writing a Polnomial Function with Comple Roots Write the simplest polnomial function with zeros i, 2 and 0. According to the Comple Conjugate Root Theorem, if a bi is a root of a polnomial equation with real-number coefficients, then There are four roots:,, 2 and 0. is also a root. P() ( i )( )( ) Write the equation in factored form. ( 2 i 2 )( )() ( 2 )( 2 ) Multipl and simplif. P() Holt Algebra 2
11 Read To Go On? Skills Intervention 6-7 Investigating Graphs of Polnomial Functions Using Graphs to Analze Polnomial Functions Identif whether the function graphed has odd or even degree and a positive or negative lead coefficient. A polnomial with an odd degree and a leading coefficient a 0: As, P () and, P () A polnomial with an even degree and a leading coefficient a 0: As, P () and, P () A polnomial with an odd degree and a leading coefficient a 0: As, P () and, P () A polnomial with an even degree and a leading coefficient a 0: As, P () and, P () Using the given graph: As, P () and, P () P() is of degree with a leading coefficient. Graphing a Polnomial Function Graph the function f() Step When p, q, possible roots are:,,, and. Step 2 Using snthetic division test possible rational zeros until a zero is identified. Test Test is a zero. So f () ( 2)( 2 ). Step 3 Factor: ( 2)( )( ) The zeros are 2,, and. Step 4 To determine the -intercept substitute for. f (0) Step 5 Plot the zeros and -intercept. Plot points between the zeros. Choose 3 and. f ( 3) ( ) 8 Plot the ordered pair ( 3, ). f ( ) ( ) 8 Plot the ordered pair (, ). Step 6 Identif the end behavior: As, P() and, P(). Step 7 Sketch the graph of f () Holt Algebra
12 Read To Go On? Skills Intervention 6-8 Transforming Polnomial Functions Reflecting Polnomial Functions Let f () Write a function g () that performs each transformation. A. Reflect f () across the -ais. When ou reflect a function across the -ais ou take the g () f () g () ( ) g () Check: Graph both functions on a graphing calculator. Do our graphs appear to be reflections? B. Reflect f () across the -ais. of f (). When ou reflect a function across the -ais ou take the opposite of, f ( ). g () f ( ) g () ( ) 4 8 g () What do ou notice about both functions? Compressing and Stretching Polnomial Functions Let f () Graph f and g on the same coordinate plane. Describe g as a transformation of f. A. g () 2f() g () ( ) Multipl each term b 2. g () g () is a stretch of f (). f () is graphed on the grid. Graph g () on the same grid. B. g () f 2 g () g () g () is a horizontal of f (). f () is graphed on the grid. Graph g () on the same grid Holt Algebra 2
13 Read To Go On? Skills Intervention 6-9 Curve Fitting with Polnomial Models Using Finite Differences to Determine Degrees Use finite differences to determine the degree of the polnomial that best describes the data. A The -values increase b a constant differences of the -values.. Find the To find the differences in the -value subtract each -value from the -value that follows it. For eample: 8 ( 47) First differences: 39 3 Are the differences constant? Second differences: 26 Are the differences constant? Third differences: 8 Are the differences constant? The differences are constant. Use the table above to determine the tpe of polnomial that best describes the data. A polnomial best describes the data. Use the cubic regression feature on our calculator to determine the polnomial. f () 2 B The -values increase b a constant Find the differences of the -values. First differences: 4 5 Are the differences constant? Second differences: 36 Are the differences constant? Third differences: 30 Are the differences constant? Fourth differences: Are the differences constant? The differences are constant. Use the table above to determine the tpe of polnomial that best describes the data. A polnomial best describes the data. Use the f () 2 Finite Differences of Polnomials Function Tpe Degree regression feature on our calculator to determine the polnomial. Constant Finite Difference Linear First Quadratic 2 Second Cubic 3 Third Quartic 4 Fourth Quintic 5 Fifth 03 Holt Algebra 2
14 Read To Go On? Problem Solving Intervention 6-9 Curve Fitting with Polnomial Models To create a mathematical model for the given data, ou will need to determine what tpe of function is most appropriate. Often, real-world data can be too irregular for ou to use finite differences or find a polnomial function that fits perfectl. In these situations, ou can use the regression feature on our graphing calculator. The table shows the population of wild turkes released into the wild at a state forest. Write a polnomial function for the population. Time (ears) Number of Wild Turkes Understand the Problem. What are ou being asked to do? Make a Plan 2. In order to find the of the polnomial function, make a scatter plot. Solve 3. Make a scatter plot of the data. Let represent the number of ears since the release. 4. Use the regression feature on our graphing calculator to check the R 2 values. Quadratic R Cubic R Quartic R 2 Remember that the closer the R 2 -value is to, the better the function fits the data. 5. Which function is the most appropriate? 6. Write the polnomial model: f () Look Back 7. Check one of the points in the model in Eercise 6. Tr 2. f (2) f (2) 8. Did ou get a value near 66? 04 Holt Algebra 2
15 Read To Go On? Quiz 6-5 Finding Real Roots of Polnomial Equations. The earl profit of a compan, in thousands of dollars, can be modeled b P(t ) t 4 3 t 2 36, where t is the number of ears since Factor to find the ears in which the profit was 0. Identif the roots of each equation. State the multiplicit of each root Fundamental Theorem of Algebra Write the simplest polnomial function with the given roots. 5., 2, i, 2, 7. Solve b finding all roots. 6-7 Investigating Graphs of Polnomial Functions Graph each function. 8. f () f () Holt Algebra 2
16 Read To Go On? Quiz continued Identif whether the function graphed has an odd or even degree and a positive or negative leading coefficient Transforming Polnomial Functions Let f () Write a function g () that performs each transformation. 3. Reflect f () across the -ais. 4. Reflect f () across the -ais. Let f () Graph f () and g () on the same coordinate plane. Describe g () as a transformation of f (). 5. g () 2f () 6. g () f() 7. g () f ( 2) Curve Fitting with Polnomial Models 8. The table shows the population of a bacteria Time (h) colon over time. Write a polnomial function Number of for the data. bacteria 06 Holt Algebra 2
17 Read To Go On? Enrichment Packaging A bo with a lid is to be made b cutting a 24-inch b 36-inch piece of cardboard along the dotted lines as shown in the figure and folding the flaps up along the sides. What is the maimum volume of the bo? What value of will produce a bo of maimum value? What are the dimensions of the bo? Determine the dimensions of the bo. 2. Write an equation for the volume of the bo. 3. Use the Table feature on a graphing calculator with Tbl to fill in the chart, where Y is the volume. X Y For what interval of does the maimum volume appear to eist? 5. Reset the table b setting TblStart to the lower value of the interval in Eercise 4 and Tbl to 0.. In what interval of does the maimum volume appear to eist? 6. Reset the table b setting TblStart to the lower value of the interval in Eercise 5 and Tbl to What appears to be the value of that produces the maimum volume? 7. What is the maimum volume of the bo? 8. What are the dimensions of the bo? 07 Holt Algebra 2
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