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Karo-lina-s [1.5K]
3 years ago
12

How many cubic (i.e., third-degree) polynomials $f(x)$ are there such that $f(x)$ has positive integer coefficients and $f(1)=9$

? (note: all coefficients must be positive---coefficients are not permitted to be 0, so for example $f(x) = x^3 + 8$ is not a valid polynomial.)?
Mathematics
1 answer:
Firdavs [7]3 years ago
6 0
Hello,
there are 56 polynomials : a*x^3+b*x^2+c*x+d=0
num            a  b  c  d
 1             1  1  1  6
 2             1  1  2  5
 3             1  1  3  4
 4             1  1  4  3
 5             1  1  5  2
 6             1  1  6  1
 7             1  2  1  5
 8             1  2  2  4
 9             1  2  3  3
 10            1  2  4  2
 11            1  2  5  1
 12            1  3  1  4
 13            1  3  2  3
 14            1  3  3  2
 15            1  3  4  1
 16            1  4  1  3
 17            1  4  2  2
 18            1  4  3  1
 19            1  5  1  2
 20            1  5  2  1
 21            1  6  1  1
 22            2  1  1  5
 23            2  1  2  4
 24            2  1  3  3
 25            2  1  4  2
 26            2  1  5  1
 27            2  2  1  4
 28            2  2  2  3
 29            2  2  3  2
 30            2  2  4  1
 31            2  3  1  3
 32            2  3  2  2
 33            2  3  3  1
 34            2  4  1  2
 35            2  4  2  1
 36            2  5  1  1
 37            3  1  1  4
 38            3  1  2  3
 39            3  1  3  2
 40            3  1  4  1
 41            3  2  1  3
 42            3  2  2  2
 43            3  2  3  1
 44            3  3  1  2
 45            3  3  2  1
 46            3  4  1  1
 47            4  1  1  3
 48            4  1  2  2
 49            4  1  3  1
 50            4  2  1  2
 51            4  2  2  1
 52            4  3  1  1
 53            5  1  1  2
 54            5  1  2  1
 55            5  2  1  1
 56            6  1  1  1

DIM a AS INTEGER, b AS INTEGER, c AS INTEGER, d AS INTEGER, k AS INTEGER
OPEN "c:\nosdevoirs\polynome.sol" FOR OUTPUT AS #1
k = 0
FOR a = 1 TO 6
    FOR b = 1 TO 6
        FOR c = 1 TO 6
            FOR d = 1 TO 6
                IF a + b + c + d = 9 THEN
                    k = k + 1
                    PRINT k, a; b; c; d
                    PRINT #1, k, a; b; c; d

                END IF
            NEXT d
        NEXT c

    NEXT b
NEXT a
CLOSE #1
END



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a₁r^{4} = 162 → (1)

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Divide (2) by (1)

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     = 59048

   

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