5.4 FACTORING AND SOLVING POLYNOMIAL EQUATIONS To help you with #1-1 THESE BINOMIALS ARE EITHER GCF, DOS, OR BOTH!!!! Just GCF Just DOS Difference of Perfect Squares Both 1. Break each piece down.. Pull out what they have in common and put it in front of the( ). What s left over stays inside the parenthesis. 7a 6a 1. Determine the each part is a perfect square and that it is subtraction!! Remember a perfect square is the result of something times itself.. Break down each part into what makes it a perfect square.. Put a minus in one ( ) and a plus in the other. x 4 64 1. Pull out the GCF.. You notice that s what in the ( ) is the difference (subtraction) of two perfect squares. 5x 4 5 x 4 605 11 Now the directions say solve or find the real number solutions : 9 a 0 a 4 0 x 8 0 x 8 0 x 11 0 x 11 0
To help you with #1- THESE TRINOMIALS MIGHT HAVE A GCF!! Just TRINOMIAL TRINOMIAL WITH GCF 1. Break down 4 x. Look for numbers that multiply to -48 and add to +.. Check each parenthesis to make sure you can t factor any further. One of the parentheses you create could be a DOS. x 4 x 48 1. Pull out the GCF.. Break down 4 x. Look for numbers that multiply to 1 and add to -7. 4. Check each parenthesis to make sure you can t factor any further. One of the parentheses you create could be a DOS. x 4 1x 6 x 4 7x 1 Now the directions say solve or find the real number solutions :
To help you with #-9 these BINOMIALS are either Sum of Cubes or a Difference of Cubes! Be careful, there might be a GCF hiding the real problem!! Sum of Two Cubes Difference of Two Cubes (a + b)(a ab + b ) (a b)(a + ab + b ) Numbers that are perfect cubes: 1, 8, 7, 64, 15, 16, 4, x, a Without GCF 1. Break down each term.. Whatever is being cubed first is called your a value.. Whatever is being cubed second is called your b value. 4. Put a and b into the appropriate formula. x 7 With GCF 1. Pull out the GCF.. Break down each term.. Whatever is being cubed first is called your a value. 4. Whatever is being cubed second is called your b value. 5. Put a and b into the appropriate formula. 4x 8x 1 To help you with #0-4 terms should be factored by GROUPING!! 1. Make two groups.. Pull out the GCF of each group. Notice, in the second group, - 4 had to be pulled out so that the two remaining parentheses are identical.. Pull out the common ( ) of x 5 4. Whatever is leftover creates its own ( ). 5. Check your answer to see if it needs to be factored further!! x 5x Groups : GCF : 4x 0
5.5 DIVIDING POLYNOMIALS WE HAVE TWO CHOICES: LONG DIVISION OR SYNTHETIC DIVISION. Think about how you would normally do long division 0 450 Example 1: Dividing polynomials using long division. Make sure that the polynomial is in descending order (standard form). If one of the terms is missing, you must put a placeholder of 0 in its place. f x x x 18 x x x x 18 4 h x 4x 5x 9x 18 x x 4
Example : Dividing polynomials using synthetic division. Where have we heard synthetic before?? To divide using synthetic division, you must also make sure that each part of the problem is in standard form. If not, fill in the missing terms with 0. You will be given the factor to divide by. For synthetic division, we turn factors into zeros first by setting them equal to 0 and solving for x. x 4x 5x x Notice that we are dividing by a factor x +. In synthetic division, we must turn that factor into its zero form by setting it equal to zero and solving. x 0 x After you divide you will see the coefficients of your answer below the line. Look back to the original problem. Our answer is always one power smaller than the original. Since our original was x, our answer will descend from x. 1 4 5 4 x 5x 8x 1 x 6 x 15 x 5
5.5 USING THE FACTOR THEOREM TO FIND THE ROOTS OF A POLYNOMIAL The Factor Theorem: Forget polynomials for a minute and think back to elementary school when you only worked with integers. How did you know that 4 is a factor of or goes into 4? Can you prove it? How? Switch back to polynomials: How can you prove that x is or is not a factor of f ( x) x 11x x 6? You can prove that x is a factor of the given polynomial by: If, then. Try it: We now know that x is definitely a factor of f ( x) x 11x x 6. Look at what s left after we divided out the x. How could you use what s left to find all of the remaining factors that make up the original polynomial??? List all of the factors of f ( x) x 11x x 6 :
g and one of its factors 5 4 Example: The polynomial ( x) 4x 8x 81x 16 completely. x are given. Use it to factor 5 4 List all of the factors of g ( x) 4x 8x 81x 16 : Example: One zero of f ( x) x x 71x 0 is x = 6. Find the other zeros. (What s the difference between a factor and a zero?) List all of the zeros of f ( x) x x 71x 0 : Example: One zero of f ( x) x 6x 7x 0 is x =. Find the other zeros. List all of the zeros of f ( x) x 6x 7x 0 :
5.6 FINDING RATIONAL ZEROS Fundamental Theorem of Algebra Every polynomial equation with degree greater than 0 has at least one zero (spot where the graph crosses the x-axis). That zero can be one of three types: a positive real number, a negative real number, or an imaginary number. Find all of the real zeros of p ( x) x 4x 7x 10 Identify p (the constant) and identify q (the leading coefficient) List all the factors of p and q Make the p list (the list of possible rational zeros): taking one at a time, divide each q factor of p by each factor of q. This is the list of all possible rational zeros. Once the list is created, graph the polynomial on your calculator to figure out which possible zeros should actually be tested. Test out the estimated zeros using synthetic division. Once you get one that works (remainder of 0), factor what remains and turn each factor into its zero form. p q Find all of the real zeros of: 1. p ( x) x 4x 7x 10 How many zeros should we expect to get?. g ( x) x 1x 0 How many zeros should we expect to get? p q
4. p ( x) x 5x 5x 5x How many zeros should we expect to get? How many times will we need to do synthetic division? Why? 4 4. g( x) 4x 15x 88x 1x How many zeros should we expect to get? How many times will we need to do synthetic division?
5.7 Apply the Fundamental Theorem of Algebra Fundamental Theorem of Algebra In other words, every polynomial equation with degree greater than 0 has at least one zero/solution (spot where the graph crosses the x axis). That zero can be one of three types: a positive real number, a negative real number, or an imaginary number. Specifically, if you know the degree, you know exactly how many zeros/solutions the polynomial has. Find the number of solutions or zeros. a. Because x x + 9x 7 = 0 is a degree polynomial function, it has solutions or zeros. b. Because x 4 + 6x x = 5 is a degree polynomial function, it has solutions or zeros. Find all the zeros of: (What word is missing from yesterday??? What might that mean about today s answers?) 4 1. f ( x) x 7x 1x x 0 Because the function is a function, it has zeros. f ( x) x 4 7x 1x x 0 With the p/q list in mind, check the calculator for zeros that should be tested out. Use synthetic division to divide out the known zeros. Factor what remains. If factoring is not possible, use the quadratic formula to get the remaining zeros. b x b 4a c a
5 4. p ( x) x 5x 9x 5x 8x 1 Because the function is a degree polynomial function, it has zeros. With the p/q list in mind, check the calculator for zeros that should be tested out. Use synthetic division to divide out the known zeros. Factor what remains. If factoring is not possible, use the quadratic formula to get the remaining zeros. b x b 4a c a Complex Conjugates Theorem Irrational Conjugates Theorem In other words, complex conjugates and irrational conjugates occur in pairs. If one of them is a zero of the function, then the other is also a zero of the function.
Write a polynomial function f of least degree that has real coefficients, a leading coefficient of 1, and and i as zeros. x x i x Write each zero as its factored form. Multiply the parentheses that contain i. Expand, use i = 1. Simplify. Multiply. Combine like terms. Write a polynomial of least degree that has rational coefficients, a leading coefficient of 1, and 0 and 6 as zeros. x 0 x 6 x
5.8 Analyze Graphs of Polynomial Functions ZEROS, FACTORS, SOLUTIONS, AND INTERCEPTS How are they all connected? Let f(x) be a polynomial function. If you know that one of the zeros of this polynomial is x =, then you immediately know: Factor: is a factor of the polynomial f(x). Solution: is a solution of the polynomial equation f(x) = 0. x-intercept: is an x-intercept of the graph, where the graph crosses the x-axis. The graph of f contains (, ). Example: Use the x-intercepts to graph the function. f x.5x 1 x 4 1. Use the factors to find the zeros. x 1 0 x 1 0 x x x 4 0 x Turn the zeros into their x-intercept form to create points for the graph. x-intercepts:,,,. Use the table in your calculator to get additional points around the intercepts. x - - 0 5 y. Determine the end behavior. Because f has three factors of the form x k, and a constant factor of, it is a cubic function. As x, f ( x) As x, f ( x)
Turning Points Local maximum Local minimum The y-coordinate of a turning point if the point is higher than all nearby points. The y-coordinate of a turning point if the point is lower than all nearby points. Go back to the problem we just graphed. What were these turning points? Graph the function. Identify the x-intercepts and the points where the local maximums and local minimums occur. f(x) = x + 8x Before graphing the function in the calculator, determine the degree,. Because the degree is, we know that this graph can have at most turning points. Use a graphing calculator to graph the function. To change the WINDOW, hit the window key and adjust the max and min numbers. To get back to a normal screen, hit Zoom, then 6. Notice that the graph of f has x-intercepts and turning points. Identify the type of turning points that you see: Use the graphing calculator's zero, maximum, and minimum features (Under the nd TRACE menu) to approximate each. The x-intercepts of the graph are (, ) (, ) (, ) ND CALC, :zero, Left Bound?, ENTER, Right Bound?, ENTER, Guess?, ENTER The function has a local maximum at (, ) and a local minimum at (, ) ND CALC, : minimum OR 4:maximum, Left Bound?, ENTER, Right Bound?, ENTER, Guess?, ENTER