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AP CALCULUS Review 5: Long Division and Completing the Square Polynomial Long Division 1. Divide 467 12 Steps 1) Divide 2) Multiply 3) Subtract 4) Bring Down 5) Repeat 2. x 3 7 x 2 7 x 6 x 2 Step1:Divide.We will divide the first term of the polynomial by x 2. x 3 7 x 2 7 x 6 x 2 Step 2: Multiply. We multiply the piece we just put as part of the answer x 2 by the entire binomial x 2 . This is written underneath the original polynomial (just like we would in an arithmetic long division problem. 2. x 3 7 x 2 7 x 6 x 2 Step 3: Subtract. We now need to subtract x 3 2 x .Remember that to subtract a polynomial you have to change the sign of each term then combine like terms as shown here: 2. x 3 7 x 2 7 x 6 x 2 Step 4: Bring down. Simply bring down the next term in the polynomial: 2. x 3 7 x 2 7 x 6 x 2 Step 5: Repeat. Now we start back again at the beginning. Divide: Now we will divide the first term of our answer 5x2 by x., so 5x goes at the top as part of our answer: 2. x 3 7 x 2 7 x 6 x 2 Multiply: Now we multiply the piece we just put as part of the answer (5x) by the entire binomial (x+2). This is written underneath (just like we would in an arithmetic long division problem). 2. x 3 7 x 2 7 x 6 x 2 Subtract: We now need to subtract 5 x 2 10 x Remember that to subtract a polynomial you have to change the sign of each term then combine like terms as shown here: 2. x 3 7 x 2 7 x 6 x 2 5 x 2 10 x Bring Down: Simply bring down the next term in the polynomial: 2. x 3 7 x 2 7 x 6 x 2 Repeat: Once again we start back at the beginning with division Divide: Now we will divide the first term of our answer -3x by x. 3 x x 3 so -3 goes at the top as part of our answer: 2. x 3 7 x 2 7 x 6 x 2 Multiply: Now we multiply the piece we just put as part of the answer (-3) by the entire binomial (x+2).3(x + 2) = -3x -6. This is written underneath the original polynomial (just like we would in an arithmetic long division problem. 2. x 3 7 x 2 7 x 6 x 2 Step 6: Subtract. We now need to subtract -3x-6. Remember that to subtract a polynomial you have to change the sign of each term then combine like terms as shown here: Note: no remainder so we have our answer x 2 5 x 3 Divide using Polynomial Long Division 3. 2 x 8 x 11x 5 x 3 3 2 Completing the Square The process of making a quadratic expression into a perfect square by adding half the coefficient of the x term squared is called completing the square. Completing the square The process by which a quadratic can factor 4. 2 5 x 10 x 2 x 5 5. 6. 2 x 2 8x 4 2 2 x 4 2 x 2 20 x 10 x 10 2 Complete the square to find the vertex form. 7. f ( x) x 10 x 24 2 Graph, identify vertex Complete the square to find the vertex form 2 8. f ( x) 2 x 8x 7 Complete the square to find the inverse trig form 8. 1 f ( x) 2 x 4 x 10 Assignment Review Assignment 5