13 years.
Step-by-step explanation:
Suppose, today, the Johnson twins' age is x years. Then the age of their elder sister will be (x + 5) years old. It is given that product of the three siblings' ages is exactly 2998 more than the sum of their ages. Therefore:
x + x + x + 5 + 2998 = (x)(x)(x + 5).
3x + 3003 = x^3 + 5x^2.
x^3 + 5x^2 - 3x - 3003 = 0.
To solve the question, factor theorem will be used. It states if any number c satisfies the equation f(c) = 0, then c is the factor of f(x). By observation, it can be seen that x = 13 can be a possible candidate for the factor of f(x). To verify, check f(13).
f(13) = 13^3 + 5(13^2) - 3(13) - 3003 = 2197 + 845 - 39 - 3003 = 0.
Since f(13) = 0, therefore x = 13 is the factor of f(x).
Now applying synthetic division:
Column 1 2 3 4 5
13 | 1 5 -3 -3003 (Row 1)
| _13__234___3003 (Row 2)
| 1 18 231 0 (Row 3)
The method requires to write the coefficients of all the powers in the Row 1 on the right hand side of the table. Put the factor x = 13 on the left hand side of the table (Row 1 Column 1). The first step involves the second column elements to be added. Since there is only 1, thus there will be 1 below the line. Next step involves multiplying 1 with the factor (which is 13) and this will be the element for row 2 column 3. The add the elements of the column 3. The answer comes out to be -3. Repeat the above steps for the columns 4 and 5. The sum of the column 5 (the last column) should be 0, which is the case. The elements of the last row are actually the coefficients of the quadratic equation which is resultant from dividing f(x) by (x - 13). Thus:
x^2 + 18x + 231 = 0.
Using completing the square:
x^2 + 18 = -231.
(x)^2 + 2*(x)*(9) + (9)^2 = -231 + (9)^2.
(x + 9)^2 = -231 + 81.
(x + 9)^2 = -150.
Taking square root on both sides shows that rest of the 2 answers of this polynomial will be complex roots since square root of -150 does not exist in real numbers. Therefore, since the age is a real quantity and not a complex quantity, the age of the twins is 13 years!!!
