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Polynomial factors and graphs — Basic example

Watch Sal work through a basic Polynomial factors and graphs problem.

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  • area 52 blue style avatar for user PC8923
    Why is it that the examples in these videos are always so easy but the practice problems are so much harder?
    (63 votes)
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  • leafers ultimate style avatar for user Asad Ali
    am i the only one who tried it with qudratic formula
    and it still worked!
    (12 votes)
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  • blobby green style avatar for user 25rodriguez
    wouldn't zeroes just generally mean all intercepts?
    (7 votes)
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  • boggle blue style avatar for user Jan
    all the topics are not covered so its hard to understand the quizzes
    (8 votes)
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  • blobby green style avatar for user Mersi
    I encountered a problem where I needed to find the minimum x-coordinate of the parabola. I forgot the formula and I used the calculator since it was in the calculator section. I got an x = 0.4999999999. With the formula, it was 0.5. Would the answer be marked wrong on the SAT test if I put .499 and not 0.5(as the result from the formula)?
    (4 votes)
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    • piceratops ultimate style avatar for user Hecretary Bird
      I believe that it would be marked wrong. Equivalent forms of the expression are permitted, and decimal approximations of repeating fractions or fractions that aren't repeating but can't be represented exactly with the three digits you get, but none of those apply to this. I guess unless the question has some inequality in it, you should automatically assume that 0.49999 means .5.
      (7 votes)
  • male robot hal style avatar for user Tyler
    The best thing so far is the SAT has only one right option in the questions. There is no " choose the one which is more correct". So, carry on comrades. Our goals are gonna be achieved.
    (3 votes)
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  • blobby green style avatar for user Sophia Lowden
    What is the definition of a polynomial? I tried looking it up but I don't understand what google gives me.
    (1 vote)
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    • piceratops ultimate style avatar for user Hecretary Bird
      A polynomial is a string of terms. These terms each consist of x raised to a whole number power and a coefficient. As an example, take the polynomial 4x^3 + 3x + 9. Since this has three terms, it's called a trinomial. Two-term polynomials are binomials and one-term polynomials are monomials. The 9 term would technically be multiplied to x^0, but since that is 1 we leave it out. We'd call this polynomial a 3rd-degree polynomial because its highest exponent of x is 3. The roots or zeroes of a polynomial are its x-intercepts when it's plotted on a graph.
      The two main rules of polynomials are that you can only have one variable, and that only whole number powers are allowed. Polynomials don't have negative powers, because then you would have a variable in the denominator. Fractional powers also don't count.
      We can use polynomials to model real life situations, and they all have predictable graphs. Odd-degree polynomials will start from the top/bottom of the graph on the -x side and end up at the opposite side on the +x side of the graph. Even-degree polynomials will look like a "u": As x goes towards negative infinity, the y value will either be extremely positive or negative, and it will be on the same side for when x tends towards positive infinity.
      Hopefully this is a good enough starting point.
      (5 votes)
  • blobby green style avatar for user Rayeed
    wont we check the number of turning/stationary points? What type of question will we be asked in order to check that?
    (2 votes)
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  • blobby green style avatar for user yeona kim
    how to solve these problems when graph is not givin?
    (1 vote)
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    • piceratops ultimate style avatar for user Hecretary Bird
      The graph is your answer choice here. If you're asked to identify a polynomial with 4 distinct zeroes and given an equation for example, the correct answer would be able to be split into 4 different factors that look like (x-r), with obviously different numbers for the r-value.
      The root form of a polynomial, y = a(x - r1)(x - r1)... gives you all the roots like that, in factored form.
      (1 vote)
  • leafers ultimate style avatar for user Mathew
    Would you need to factor the problem out, it seemed like a trick question?
    (1 vote)
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Video transcript

- [Instructor] The polynomial p of x has four distinct zeros, four distinct zeros. Which of the following graphs could represent y is equal to p of x. Well, four distinct zeros means that the graph is going to intersect the x-axis exactly four times. So this first one intersects the x-axis one, two, three times. So we can rule that one out. This one intersects the x-axis one, two, three, four, five times. Let's go and rule that one out. This one intersects the x-axis one, two, three, four times. It has four distinct zeros. So this looks like, this one right over here looks like our choice, and this one right over here, it actually only has two distinct zeros, or at least that's what it looks like. So I'd rule that one out as well. So this is definitely going to be the one that has four distinct zeros. Zeros are the x values at which the function is equal or the polynomial, in this case, which is a function of x, is going to be equal to zero, and you can see that's happening exactly four times.