To determine which expression represents a dimension of Vincent's rectangular poster, we first need to factor the area expression \( x^2 - 17x - 60 \).
To factor \( x^2 - 17x - 60 \), we need to find two numbers that multiply to \(-60\) (the constant term) and add to \(-17\) (the coefficient of the linear term).
The pairs of factors of \(-60\) are:
- \(1, -60\)
- \(-1, 60\)
- \(2, -30\)
- \(-2, 30\)
- \(3, -20\)
- \(-3, 20\)
- \(4, -15\)
- \(-4, 15\)
- \(5, -12\)
- \(-5, 12\)
- \(6, -10\)
- \(-6, 10\)
Among these pairs, the numbers \(-20\) and \(3\) multiply to \(-60\) and add to \(-17\).
Thus, we can factor the expression as follows:
\[ x^2 - 17x - 60 = (x - 20)(x + 3) \]
Now, we can see that the dimensions of the poster can be represented by the factors \( (x - 20) \) and \( (x + 3) \).
Reviewing the answer choices, we find:
- \((x - 3)\)
- \((x + 12)\)
- \((x - 20)\)
- \((x - 5)\)
Among these, \((x - 20)\) is the only factor we found when we factored the area.
Therefore, a dimension of the poster is represented by \((x - 20)\).
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