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Basic About Polynomials

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Published in: Mathematics
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This presentation includes some basic knowledge about Polynomials for students up to 10th Grade. This does not include all the details but to some extent it will provide basic idea bout topic

Umer H / Sharjah

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  1. TOPIC POLYNOMIALS PRESENTED BY: UMER HASSAN
  2. Remember integers are . 1, O, 1, 2 ... (no decimals or fractions) so positive integers would be O, 1, 2 . polynomial function is a function of the form: n must be a positive integer All of these coefficients are real numbers The deg of the polynomial is the largest power on any xterm in the polynomial.
  3. Determine which of the following are polynomial functions. If the function is a polynomial, state its degree. A polynomial of degree 4. We can write in an xo since this = 1 u A polynomial of degree O. Not a polynomial because of the square root since the ower is NOT an integer 3 Not a polynomial because of the x in the denominator since t e power is negative
  4. Graphs of polynomials are smooth and continuous. N har corners or cus without lifting pencil from paper This IS the graph of a polynomial This •S NOT the graph of a polynomial
  5. Let's look at the graph of f (X) = x n where n is an even integer. 4 h(x) = x 6 2 Notice each graph looks similar to but is wider and flatter near the origin between —1 and 1 and grows steeper on either Side The higher the power, the flatter and steeper
  6. Let's look at the graph of f (X) integer. Notice each graph looks similar to but is wider and flatter near the origin between —1 and 1 5 3 X where n is an odd and grows steeper on either side h(x) = x 7 The higher the power, the flatter and steeper
  7. Let's graph f (X) Reflects about the x-axis Translates up 2 Looks like but wider near origin and steeper after I and -1 So as long as the function is a transformation of M', we can graph it, but what if it's not? We'll learn some techniques to help us determine what the graph looks like in the next slides.
  8. and RIGHT HAND BEHAVIOUR OF A GRAPH of the polynomial along with the sign of the The deg ree coefficient of the term with the highest power will tell us about the left and right hand behaviour of a graph.
  9. Even degree polynomials rise on both the left and hand sides of the graph (like if the coefficient is positive. The additional terms may cause the graph to have some turns near the center but will always have the same left and right hand behaviour determined by the highest powered term. left hand behaviour: rises right hand behaviour: rises
  10. Even degree polynomials on both the left and hand sides of the graph (like - if the coefficient is negative. turning points in the middle left hand behaviour: falls right hand behaviour: falls
  11. Odd degree polynomials and rise on the left on hand sides of the graph (like if the the coefficient is positive. turning Points in the middle left hand behaviour: falls right hand behaviour: rises
  12. Odd degree polynomials rise on the left and on hand sides of the graph (like if the the coefficient is negative. turning points in the middle left hand behaviour: rises right hand behaviour: falls
  13. A polynomial of degree n can have n-l turning points (so whatever the degree is, subtract 1 to get the most times the graph could turn). Let's determine left and right hand behaviour for the graph of the function: xo f X — 4 —15X2 -k 19X+30 degree is 4 which is even and the coefficient is positive so the graph will look like x2 looks off to the left and off to the right. The graph can have at most 3 turning points How do we determine what it looks like near the middle?
  14. x and y intercepts would be useful and we know how to find those. To find the y intercept we put O in for x. f (O) = 04 — — -k 30 = 30 To find the x intercept we put O in for y. Finally we need a smooth curve through the intercepts that has the correct left and right hand behavior. To pass through these points, it will have 3 turns (one less than the degree so that's okay) Y -40
  15. We found the x intercept by putting O in for (x) or y (they are the same thing remember). So we call the x intercepts the zeros of the polynomial since it is where it = O. These are also called the roots of the polynomial. Can you find the zeros of the polynomial? There are repeated factors. (x-1) is to the 3rd power so it is repeated 3 times. If we set this equal to zero and Ive we get 1. We then say that 1 is a zero of multiplicity (since it showed up as a factor 3 times). zeros and their multiplicities? -2 is a zero of multiplicit 3 is a zero of multiplicity 1
  16. So knowing the zeros of a polynomial we can plot them on the graph. If we know the multiplicity of the zero, it tells us whether the graph crosses the x axis at this point (odd multiplicities CROSS) or whether it just touches the axis and turns and heads back the other way (even multiplicities TOUCH). Let's try to graph: What would the left and right hand behavior be? ultiply this out but figure out what the You don't nee highest power on an x be if multi lied out. In this case it would be an Notice ne ative out in front. What would the y intercept be? Find the zeros and their multiplicity 1 of mult. 1 (s crosses axis -2 of mult. 2 (so touches at 2)
  17. Steps for Graphing a Polynomial •Determine left and right hand behaviour by looking at the highest power on x and the sign of that term. 'Determine maximum number of turning points in graph by subtracting 1 from the degree. •Find and plot y intercept by putting 0 in for x 'Find the zeros (x intercepts) by setting polynomial = 0 and solving. 'Determine multiplicity of zeros. •Join the points together in a smooth curve touching or crossing zeros depending on multiplicity and using left and right hand behavior as a guide.
  18. Let's graph: 'Join the points together in a smooth curve touching or crossing zeros depending on multiplicity and using left and right hand behaviour as a guide. Here is the actual graph. We did pretty good. If we'd wanted to be more accurate on how low to go before turning we could have plugged in an x value somewhere between the zeros and found the y value. We are not going to be picky about this though since there is a great method in calculus for finding these maxima and minima. Y -.30
  19. What is we thought backwards? Given the zeros and the degree can you come up with a polynomial? Find a polynomial of degree 3 that has zeros —1, 2 and 3. What would the function look like in factored form to have the zeros given above? Multiply this out to get the polynomial. FOIL two of them and then multiply by the third one. f (x) = x3 — 4x2 + 6

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