1 / 9

Zeros

Zeros. Zero to hero! …or was it Gyro?. From Yesterday. How do I tell how many zeros a function has? It’s the same as the degree of the function. How do we find the zeros?. The long way –make a t-chart For our homework, only go from -4 to +4

shalom
Download Presentation

Zeros

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Zeros Zero to hero! …or was it Gyro?

  2. From Yesterday • How do I tell how many zeros a function has? • It’s the same as the degree of the function

  3. How do we find the zeros? • The long way –make a t-chart • For our homework, only go from -4 to +4 • Any time two consecutive values on a t-chart go from positive to negative, you have a zero in between them • To approximate zeros, repeat the t-chart process using tenths-place values

  4. Example • f(x) = x4 + 3x3 – 5 • Make a T-chart • Plug in x-values from -4 to 4 • Iterate

  5. Is there a shortcut? • Kind of • First, Descartes’ Rule of Signs

  6. Descartes’ Rule of Signs • Used to determine number of types of zeros • For positive real zeros: • Count the number of times the sign on the coefficients changes • For negative real zeros: • Find f(-x), then repeat what we did for positives.

  7. One other thing to note • Imaginary roots: if you have not-imaginary coefficients, they always happen in conjugate pairs • This means there are always two, and you always just flip the sign on the i-term

  8. Now how about that shortcut? • Remember synthetic division? • f(x) = 2x3 – 17x2 + 90x – 41 • f(1/2) • The ½ is a given value of x for which the function is zero; in other words, it’s an answer. • It also means that (x – ½ ) is a binomial factor of the function. In other words you can divide f(x) by that binomial and it will come out evenly. This is why we use synthetic division here – to find what’s left when you divide.

  9. We also go the other direction • If we know all the zeros, but not the function, we can figure out the version with the smallest integer coefficients easily. • Make each zero into a binomial • i.e., if f(2)= 0, (x – 2) is the corresponding binomial. • It’s always x – (the zero) • Multiply all the binomials together (FOIL/distribution) • Example: • f(2) = 0, f(3) = 0, f(-2) = 0 • (x – 2)(x – 3)(x + 2) = f(x) • x3 – 3x2 – 4x + 12 = f(x)

More Related