A wooden beam is left parenthesis 6 y squared plus 7 y plus 1 right parenthesis6y2+7y+1 meters long. If a piece of length left parenthesis y squared minus 9 right parenthesisy2−9 meters is cut​ off, express the length of the remaining piece of beam as a polynomial in y.

Answer 1

To find the length of the remaining piece of the wooden beam after cutting off a certain piece, we need to perform a polynomial subtraction.

The initial length of the wooden beam is given by:

\[ 6y^2 + 7y + 1 \]

The length of the piece cut off is:

\[ y^2 - 9 \]

To find the remaining length, we subtract the length of the piece cut off from the original length:

\[ (6y^2 + 7y + 1) - (y^2 - 9) \]

Let's perform the subtraction step-by-step.

1. Distribute the negative sign in front of the second polynomial: \[ 6y^2 + 7y + 1 - y^2 + 9 \]

2. Combine like terms:

   • For the \(y^2\) terms: \(6y^2 - y^2 = 5y^2\)
   • For the \(y\) terms: \(7y\) (there is no corresponding \(y\) term in the second polynomial)
   • For the constant terms: \(1 + 9 = 10\)

Putting it all together, we have:

\[ 5y^2 + 7y + 10 \]

Thus, the length of the remaining piece of the beam is:

\[ \boxed{5y^2 + 7y + 10} \]