How do you factor # x^4 + x^3 + x^2 + x + 1#?
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1 Answer
Explanation:
Cheetah3d 7 3 3 X 2
This quartic has four zeros, which are the non-Real Complex #5# th roots of #1# , as we can see from:
So if we wanted to factor this polynomial as a product of linear factors with Complex coefficients then we could write:
A cleaner algebraic approach is to notice that due to the symmetry of the coefficients, if #x=r# is a zero of #x^4+x^3+x^2+x+1# , then #x=1/r# Colorsnapper2 1 6 18. is also a zero.
Hence there is a factorisation in the form:
So let's look for a factorisation:
Equating coefficients we find:
Substituting #b=-1/a# in #a+b=1# we get:
Hence:
Using the quadratic formula, we can deduce:
Since our derivation was symmetric in #a# and #b# , one of these roots can be used for #a# and the other for #b# , to find:
If we want to factor further, use the quadratic formula on each of these quadratic factors to find the linear factors with Complex coefficients.
Related questions
X-7 | |
---|---|
Role | Experimental aircraft |
Manufacturer | Lockheed Corporation |
First flight | April 26, 1951 |
Retired | 1960 |
Primary user | United States Air Force |
Developed into | AQM-60 Kingfisher |
A Lockheed X-7 on public display in New Mexico
The Lockheed X-7 (dubbed the 'Flying Stove Pipe') was an American unmanned test bed of the 1950s for ramjet engines and missile guidance technology. It was the basis for the later Lockheed AQM-60 Kingfisher, a system used to test American air defenses against nuclear missile attack.
Early Development[edit]
Development of the Kingfisher was first initiated in December 1946. The X-7 was called into production by the United States Air Force requirement for the development of an unmanned ramjet test plane with a top speed of at least Mach 3 (2300 mph).[1]
The X-7 project was developed under the AMC designator MX-883 and was given in the Lockheed in-house designation L-171. The L-171 was initially designated the PTV-A-1 by the USAF but was later designated the X-7 in 1951.[1] Despite its first launch being a failure, after re-development of the original ramjet, following test flights were successful.[2] A total of 130 X-7 flights were conducted from April 1951 to July 1960.[2]
Purpose[edit]
The X-7 laid the foundation for the AQM-60 Kingfisher.[1] Being the testbed for several yearlong projects, the X-7 underwent many structural changes to adapt more closely for its intended purpose.[1] The Kingfisher was put up against three surface to air missiles designed to test the capabilities of the X-7; SAM-A-7/MIM-3 Nike Ajax, SAM-A-25/MIM-14 Nike Hercules, and IM-99/CIM-10 Bomarc were the missiles used in the tests.[3] During the testing of the SAMs, the X-7 outperformed the missiles and a very small number of critical hits were achieved.[2] Due to the pressure and embarrassment put on the military the X-7 project was canned in the mid-1960s.[2]
Boom2:volume boost & equalizer 1 6 6. Besides the surface to air missile tests, the X-7 project was also used to test communication equipment for acceleration testing, testing aerodynamics, booster propellants, thermodynamics, and parachutes.[2]
Lockheed X-7A-1 being prepared for loading and test flight. The white section is the booster. (B-50 seen in background)
Construction[edit]
The X-7 was constructed from steel, unlike its successors such as the A-12 and YF-12 which used titanium. These [X-7] planes had wings constructed from stainless steel but had a fuselage crafted from nickel alloy. The use of steel was due to the inability of aluminum to handle the air friction at hypersonic speeds, years before the widespread introduction of titanium.
The engines developed for the X-7/AQM-60 were only designed to operate for a short time, to test the design for the CIM-10 Bomarc. They were redesigned with better materials in order to be used on the hypersonic Lockheed D-21 drone fired off the back of the Lockheed SR-71 Blackbird, or from under the wing of a Boeing B-52 Stratofortress.[4]
Launch and Recovery of the X-7[edit]
Lockheed X-7 buried nose down in the desert
The X-7 was launched at speed release from the underside of either a B-29 or B-50 carrier plane.[5] The jet would then take over and build up speed to its top speed of 1000 miles per hour, but was later redesigned to push Mach 4.3 or 2800 miles per hour.[1]
The recovery method of the X-7 rocket plane was a new and simple design for a test plane of its kind but functioned as designed. A several stage parachute was deployed after the jet had exhausted its fuel supply, slowing its descent toward the desert floor.[2] Once it had reached the floor, the long metal rod on the end of the nose cone skewered the ground keeping the plane upright and preventing any damage to structure of the X-7. In 1954, the modified X-7 underwent serious changes and was renamed the X-7A-3.[1] The wing shape was altered, and two small boosters were added to the plane; one under each wing. Due to these alterations, the drop method previously used was changed to make up for these changes. The previous version was a complicated and bulky under wing system while the new design allowed for a simple fuselage mounted dropping system. This system was used until its final flight in July 1960.[1]
Surviving aircraft[edit]
- X-7 on display at the Aviation Unmanned Vehicle Museum in Caddo Mills, Texas.[6]
X-7A on display at the Aviation Unmanned Vehicle Museum
Specifications of X-7A-1[edit]
- Length: 32 feet, 9 inches
- Wingspan: 12 feet
- Height: 7 feet
- Diameter: 20 inches
- Weight: 8000 pounds
- Speed: 2800 miles per hour (Maximum speed: Mach 4.31 (2,881 mph, 4,640 km/h)[2])
- Ceiling: 100,000 feet
- Range: 130 miles
- Booster: Alleghany Ballistics Lab. X202-C3 solid- fuel rocket; 467 kN for 4 seconds.
- Sustainer: Ramjet
Specifications of X-7A-3[edit]
- Length: 37 feet
- Wingspan: 10 feet
- Height: 7 feet
- Diameter: 20 inches
- Weight: 8000 pounds
- Speed: 2800 miles per hour
- Ceiling: 100000 feet
- Range: 130 miles
- Booster: X-7A-3/XQ-5: 2x Thiokol XM45 (5KS50000) solid-fuel rocket; 222 kN (50000 lb)
- Sustainer: Ramjet
See also[edit]
Related development
- Q-5 Kingfisher (based on the X-7 A-3)
Aircraft of comparable role, configuration, and era
References[edit]
- ^ abcdefg'Lockheed AQM-60 Kingfisher'. www.designation-systems.net. Retrieved 2017-04-09.
- ^ abcdefg'The Lockheed X-7'. www.456fis.org. Retrieved 2017-04-09.
- ^Facing the Heat Barrier: A History of Hypersonics, T. A. Heppenheimer, P.65
- ^ Goodall and Goodall 2002, p. 106.
- ^'Boeing: Historical Snapshot: B-29 Superfortress'. www.boeing.com. Retrieved 2017-04-10.
- ^'BGM-34B ATTACK & MULTI-MISSION RPV'. AUVM. Retrieved 12 October 2020.
Cheetah3d 7 3 3 X 27
External links[edit]
Cheetah3d 7 3 3 X 2
Wikimedia Commons has media related to Lockheed X-7. |
Cheetah3d 7 3 3 X 2 0cm
- 'Ramjet test missile takes off on supersonic flight.'Popular Science, pp. 142–143.
- Lockheed X7-A – X7-A on display at Aviation Unmanned Vehicle Museum outside Dallas, Texas, United States.
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