Difference between revisions of "1955 AHSME Problems/Problem 25"
Angrybird029 (talk | contribs) (Created page with "== Problem 25== One of the factors of <math>x^4+2x^2+9</math> is: <math> \textbf{(A)}\ x^2+3\qquad\textbf{(B)}\ x+1\qquad\textbf{(C)}\ x^2-3\qquad\textbf{(D)}\ x^2-2x-3\qqu...") |
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<math>x^2 + 3</math> leaves behind a remainder, and so does <math>x^2 - 3</math>. | <math>x^2 + 3</math> leaves behind a remainder, and so does <math>x^2 - 3</math>. | ||
− | In addition, <math>x + 1</math> also fails the test, and that takes down <math>x^2 - 2x - 3</math>, which can be expressed as <math>(x + 1)(x - 3)</math>. | + | In addition, <math>x + 1</math> also fails the test, and that takes down <math>x^2 - 2x - 3</math>, which can be expressed as <math>(x + 1)(x - 3)</math>. That leaves <math>\boxed{(\textbf{E})}</math> |
− | So the answer | + | ==Solution 2 (direct factorization)== |
+ | Notice the leading and constant terms are begging us to create a binomial. So | ||
+ | <cmath> | ||
+ | x^4 + 2x^2 + 9 = (x^4 + 6x^2 + 9) - 4x^2 = (x^2 + 3)^2 - (2x)^2 = (x^2 + 2x + 3)(x^2 - 2x + 3), | ||
+ | </cmath> | ||
+ | where both quadratics are irreducible (over the field of real numbers). | ||
+ | Hence none of the given options is a factor. So the answer is <math>\boxed{(\textbf{E})}</math> | ||
+ | |||
+ | ~VensL | ||
+ | |||
+ | ==See Also== | ||
+ | |||
+ | {{AHSME 50p box|year=1955|num-b=24|num-a=26}} | ||
+ | |||
+ | {{MAA Notice}} |
Latest revision as of 17:42, 21 June 2022
Problem 25
One of the factors of is:
Solution
We can test each of the answer choices by using polynomial division.
leaves behind a remainder, and so does .
In addition, also fails the test, and that takes down , which can be expressed as . That leaves
Solution 2 (direct factorization)
Notice the leading and constant terms are begging us to create a binomial. So where both quadratics are irreducible (over the field of real numbers). Hence none of the given options is a factor. So the answer is
~VensL
See Also
1955 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 24 |
Followed by Problem 26 | |
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