Saturday, September 28, 2013

SV #1: Unit F Concept 10: Given polynomial of 4th or 5th degree, find all zeroes (real and complex)

1). This is  a problem about finding all zeroes of a quartic polynomial. The polynomial we will be using is 25x^4 - 80x^3 + 58x^2 - 3. We will be combining all of concepts 3-9 in this problem. Also, we will be doing something different than what we did in those concepts because we will also be finding all irrational/ imaginary zeroes as well as the rational ones.

2). The viewer must pay attention to several things. One major thing the viewer needs to pay attention to is that the polynomial we are using doesn't have a term for x to the first degree. Thus, the viewer mustn't forget to put in a placeholder when doing synthetic division otherwise the problem can go wrong. Also, the viewer must pay attention to how to put the non factorable quadratic into factor form once the irrational zeroes are found.

Monday, September 16, 2013

SP #2: Unite E Concept 7: Graphing polynomials, including: x -int, y-int, zeroes (with multiplicities), end behavior. All polynomials will be factorable.

(1) This problem will go over how to graph a polynomial and find the characteristics stated in the above title. In this particular problem, we will be using the quartic polynomial  x^4 - 4x^3 - 12x^2. In order to complete the problem, we will be going through a number of steps, including factoring the polynomial, finding end behavior, solving for the x and y intercepts, calculating extrema and intervals of increase/decrease, and of course graphing.

(2)The reader must pay attention to detail for a little mistake could ruin the entire graph. Also, one thing the reader must be attentive to is the set window when plugging the polynomial into a graphing calculator to graph. This is because the problem results in large numbers and as a result the graph does not fit into the standard zoom.
  • First we factor out an x squared and factor the rest of the equation from there. The resulting factored equation is in the 2nd box of the picture above.
  • The end behavior part is as it is because the highest degree is 4 which is even, and the leading coefficient is positive, so the graph is an even positive meaning both ends point up.
  • To get the x- intercepts, you take each individual part with an 'x' from the factored equation and set them equal to zero. The zero of (0,0) is repeated because of the x squared so it has a multiplicity of 2.
  • For the y- intercept you just plug in zero into x from the original equation.
  • Maximums and minimums could be found using 2nd cal on your calculator and following the instructions. Next, find the intervals using the extrema.
  • Plot the zeroes and extrema on a graph. Next, figure out how the graph goes through each zero. To do this, refer to the multiplicities. A multiplicity of 1 means the graph goes straight through the zero, whereas a multiplicity of 2 means the graph bounces off the zero. Since the end behavior is even positive, both ends of the graph point up. Now draw the graph and you will be finished.

Monday, September 9, 2013

WPP #3: Unit E Concept 2: Identifying x- intercepts, y- intercepts, vertex (max/min), axis of quadratics and graphing them.

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SP #1: Unit E Concept 1 Identifying x- intercepts, y- intercepts, vertex (max/min), axis of quadratics and graphing them.

 (1) This problem is about finding various characteristics of a parabola using a quadratic function in the standard form. The characteristics we will be finding are the x- intercepts, the y- intercepts, the axis of symmetry, and the vertex (maximum or minimum). The quadratic we will be using is 2x^2 + 12x + 6.

(2) The viewer may not understand the concept if certain things go unnoticed. One of these things is that the viewer must be careful to complete the square correctly otherwise the rest of the information of the problem will be wrong. Additionally, the viewer must pay attention to the numbers meticulously because there is quite some work to do and simple mistakes can cause incorrect answers.

Step 1 includes completing the square. The first picture shows this step. We started with 2x^2 + 12x +6 = 0 and we ended up with 2(x + 3)^2 = 5.
 The Second step is shown below. In this we use what we got from step 1 to find various information about the graph by plugging numbers into both the graph form and the parent graph form of the quadratic.