Find all values of c that will make the polynomial a perfect square trinomial. Options: c = 49 c = 100 c = 144 c = 169

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 \( c \).