Exciting Applications of Polynomials Why Learning Polynomials?

What is a Polynomial? Example: Building Blocks: Monomials Parts of a Monomial: Exponent Variable Coefficient A Monomial is considered a Polynomial

Why Study Polynomials? (1) Many formulas in math, science, and everyday life are polynomials Examples: energy of mass volume of a cube body mass index Mastery of polynomials is needed for a deeper understanding of formulas

Why Study Polynomials? (2) Modern electronic devices rely on ICs (Integrated Circuits), which can have over a million components (e.g. transistors). The design and production of ICs require many complex tests to ensure the ICs do what they are designed to do. Automated testing relies on LFSRs (Linear Feedback Shift Registers). Why & how LFSRs work is explained in the theory of polynomials

Why Study Polynomials? (3) Encryption is used in military, business, and personal operations. Information are often encrypted for efficient transmission and to protect secrecy. (Cell phone calls are encrypted for these reasons.) To understand and to exploit why and how encryption work requires an elaborate study of polynomials.

Modern Electronic Devices (1) Flash Drive (1) 1.USB connector 2.USB mass storage controller device 3.Test points 4.Flash memory chip 5.Crystal oscillator 6.LED 7.Write-protect switch 8.Space for second flash memory chip

Modern Electronic Devices (1) Flash Drive (2) The mass storage controller is an ASIC (Application Specific Integrated Circuit) for directing the movement of data This and many other ASICs have embedded a LFSR for BIST (Built-in Self test), which warns the user if the ASIC is not fully functional

Modern Electronic Devices (2) Cell Phone (1) A microscopic microphone A speaker An LCD or plasma display A keyboard An antenna A battery A circuit board containing many ASICs: DSP (Digital Signal Processor) handles all the signal compression and decompression Microprocessor (and memory) handle operations of the keyboard and display

Modern Electronic Devices (2) Cell Phone (2) A cell phones use ASICs for its operations. (Ericsson phone shown uses the ASIC version of Z-80). The design and production of ASICs reply on the use of LFSRs. The calls to or from a cell phone are encrypted for efficient transmission and to protect secrecy. LFSRs are used in encryption.

Modern Electronic Devices (3) MP3 (Music Player)(1) MP3 is the name of a file format, just as PPT is for Powerpoint format MP3 players are basically just USB flash drives, and use USB connector + flash memory Music can be downloaded onto an MP3 if it is connected to a computer with supporting software like iTunes Key to this operation is compression. A CD track occupies about 12 times the storage as the same track in MP3, allowing MP3 more music or video in less space.

Modern Electronic Devices (3) MP3 (Music Player)(2) Shown is the inside of an iPod 50% space is used by the HD (SanDisk MP3 players use the more compact flash memory) Recent MP3 players play audio & video; older MP3 plays only audio MP3 uses a CPU (e.g. ARMTDM1) and/or a DSP, both are ASICs, for compressing/decompressing digital signals LFSRs and Polynomials are vital to MP3 developers

Modern Electronic Devices (4) Digital Camera (1) Takes still photographs and video digitally by recording images via an electronic image sensor Incorporated into many devices, e.g. PDAs, mobile phone, & vehicles. (The Hubble Space Telescope is basically a digital camera.) Has an image sensor that converts light into electrical charges  often a CCD (Charge Coupled Device) or a CMOS (Complementary Metal Oxide Semiconductor)

Modern Electronic Devices (4) Digital Camera (2) The amount of detail that the camera can capture is called the resolution, and it is measured in pixels. The more pixels a camera has, the more detail it can capture. Larger pictures can remain sharp. Each photosite on a sensor keeps track only of the total intensity of the light that strikes its surface. In order to get a full color image, most sensors use filtering to look at the light in its three primary colors. Once the camera records all three colors, it combines them to create the full spectrum.

Modern Electronic Devices (4) Digital Camera (3) Picture images can be stored in many devices: SmartMedia cards, CompactFlash cards, Memory Sticks, Hard disks, Writable CDs and DVDs Two features of digital images make compression possible: repetition – patterns that repeat can be reconstructed w/o loss of content irrelevancy – dicard details that the human eyes can detect Use ASICs to handle color processing, data compression, storage and housekeeping. LFSR & Polynomials are relevant!