Answer :
All values of \( c \), specifically \( c = 49, 100, 144, \) and \( 169 \), can make a polynomial a perfect square trinomial depending on the other terms of the polynomial.
Step-by-step explanation :
1. Understanding Perfect Square Trinomials :
A perfect square trinomial is one that can be written in the form:
\[ (ax + b)^2 = a^2x^2 + 2abx + b^2 \]
From this formula, the value \( c \) would be equivalent to \( b^2 \), and the coefficient of the linear term (the term with \( x \)) would be \( 2ab \).
2. Analyzing Given Options :
- Option 1 : c = 49
For \( c = 49 \), \( b^2 = 49 \) which implies \( b = \pm 7 \). The middle term would then be \( 2a(7) = 14a \).
- Option 2 : c = 100
For \( c = 100 \), \( b^2 = 100 \) which means \( b = \pm 10 \). The middle term would then be \( 2a(10) = 20a \).
- Option 3 : c = 144
For \( c = 144 \), \( b^2 = 144 \) which translates to \( b = \pm 12 \). This makes the middle term \( 2a(12) = 24a \).
- Option 4 : c = 169
For \( c = 169 \), \( b^2 = 169 \) which gives \( b = \pm 13 \). Consequently, the middle term would be \( 2a(13) = 26a \).
3. Conclusion :
Without specific details on the polynomial or its middle term, we can deduce that any of the provided options for \( c \) can result in a perfect square trinomial if the linear term of the polynomial matches the \( 2ab \) value corresponding to that \( c \).