A B737-800 CREW REPORTS DEPARTING SLC RWY 17 WHILE A REDUCED RWY LENGTH WAS IN EFFECT MAKING THE ACFT TKOF WT AND BALANCE DATA INACCURATE.

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5 MINS PRIOR TO PUSHBACK; WE RECEIVED OUR FINAL ACFT WT AND BAL DATA FROM THE GATE AGENT AS NORMAL. WE WERE EXPECTING RWY 16L FOR DEP AND LOADED THE FMS WITH THE DATA FOR THAT RWY. AFTER PUSHBACK AND TAXIING OUT; WE RECEIVED CLRNC FROM GND CTL TO HOLD SHORT OF A TXWY AND ASKED IF WE COULD ACCEPT RWY 17 AND STATED THE AMOUNT OF RWY AVAILABLE. I REFERRED TO THE FINAL WT AND BAL AND SAW THAT THE WTS WERE GOOD FOR RWY 17 SO WE ADVISED GND. AT THIS POINT THINGS WERE BUSY. I WAS DOING AN ENG XBLEED START BECAUSE OUR APU WAS NOT WORKING AND THE CAPT WAS TAXIING AND RECHKING THE NUMBERS FOR RWY 17. WE BOTH VERIFIED THAT WE HAD THE NECESSARY WT AND BAL NUMBER TO TAKE OFF ON RWY 17. AFTER TKOF; WE NOTICED MEN AND EQUIP AT THE ARR END OF THE RWY. WE WERE ACCELERATING NORMALLY AND MADE A NORMAL TKOF AND A NORMAL CLB. DURING INITIAL CLB; WE BOTH MADE COMMENTS ABOUT THE CONSTRUCTION ON THE TXWY AND DECIDED TO WAIT UNTIL CRUISE TO RECHK THE NUMBERS FOR THAT RWY. UPON FURTHER REVIEW; WE NOTICED THAT THE WT AND BAL WAS CALCULATED FOR FULL LENGTH OF RWY AND WE DID NOT NOTICE ANY NOTAMS ISSUED FOR THAT RWY. AFTER FURTHER RESEARCH; WE NOTICED THAT THE ATIS INDICATED THAT DUE TO CONSTRUCTION ON A TXWY; PART OF THE RWY WAS CLOSED AND THERE WAS 7690 FT AVAILABLE. NORMALLY; OUR FINAL WT AND BAL IS ADJUSTED FOR ANY CUTBACKS OR CLOSURES ON THE RWYS. BUT THERE WAS NO NOTAM ISSUED AND I GUESS OUR DISPATCHER WAS UNAWARE OF THE CLOSURE. IT APPEARS WE DEPARTED RWY 17 WITH INSUFFICIENT PERFORMANCE NUMBERS. CONTRIBUTING FACTORS ARE NUMEROUS: FIRST; IT WAS A VERY BUSY MORNING IN SLC. THE GND CTLR; RAMP AND TWR WERE ALL SATURATED AND IT WAS MISSED WHEN THE AVAILABLE LENGTH OF RWY WAS SAID. SECOND; WE WERE BUSY AS WELL. AS STATED ABOVE; WE WERE DOING A XBLEED START DUE TO OUR APU NOT WORKING. THIRD; ONCE WE WERE GIVEN RWY 17 FOR TKOF; WE HAD TO REPROGRAM THE FMS AND RUN CHKLIST. FACTORS THAT COULD HAVE PREVENTED THIS FROM OCCURRING INCLUDE: FIRST; LIST THE CONSTRUCTION AS A NOTAM. SECOND; ALLOW DISPATCHERS [TO BE] AWARE OF ANY CLOSURES ON THE ATIS SO THAT IT CAN BE REFLECTED ON THE WT AND BAL.