# Difference between revisions of "1983 AHSME Problems/Problem 18"

Sevenoptimus (talk | contribs) (Fixed LaTeX and formatting) |
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− | Problem | + | ==Problem== |

+ | |||

Let <math>f</math> be a polynomial function such that, for all real <math>x</math>, | Let <math>f</math> be a polynomial function such that, for all real <math>x</math>, | ||

− | < | + | <math>f(x^2 + 1) = x^4 + 5x^2 + 3</math>. |

− | For all real <math>x | + | For all real <math>x, f(x^2-1)</math> is |

− | (A) | + | <math>\textbf{(A)}\ x^4+5x^2+1\qquad |

+ | \textbf{(B)}\ x^4+x^2-3\qquad | ||

+ | \textbf{(C)}\ x^4-5x^2+1\qquad | ||

+ | \textbf{(D)}\ x^4+x^2+3\qquad | ||

+ | \textbf{(E)}\ \text{none of these} </math> | ||

− | Solution | + | ==Solution== |

Let <math>y = x^2 + 1</math>. Then <math>x^2 = y - 1</math>, so we can write the given equation as | Let <math>y = x^2 + 1</math>. Then <math>x^2 = y - 1</math>, so we can write the given equation as |

## Revision as of 16:53, 27 January 2019

## Problem

Let be a polynomial function such that, for all real , . For all real is

## Solution

Let . Then , so we can write the given equation as Then substituting for , we get The answer is therefore .