Section P.4 Addition and Subtraction of Polynomials

Section P.4 Addition and Subtraction of Polynomials Chapter P – Polynomials Like terms are monomials that contain the same variables raised to the same powers. #1- 16: Combine like terms. Write all answers in descending order. 1) 3x2 + 5x – 4 – 2x2 + 7x +8 = 3x2 – 2x2 + 5x + 7x – 4 + 8 Solution: x2 + 12x + 4 3) 4x3 + 5x – 2x3 + 8x + 4x3 – 9x = 4x3 – 2x3 + 4x3 + 5x + 8x – 9x Solution: 6x3 + 4x 5) 3x + 5x2 + 11x – 5 = 5x2 + 3x + 11x – 5 Solution: 5x2 + 14x - 5 7) 2 𝑥 3 1 1 2 1 3 + 3𝑥 − 2 + 2 1 3 Add the 3 𝑥 + 3 𝑥 then combine − 2 + 2 3 3 = 𝑥+ 2 2 Solution: x + 1 9) 4 2 𝑥 5 3 + 5 𝑥 2 + 5𝑥 − 4𝑥 + 1 Combine the x2 terms by adding the fractions, and also combine the 5x – 4x. Solution: 𝟕 𝟐 𝒙 𝟓 +𝒙+𝟏 11) 8xy + 2xy2 +3xy – 5x2y = 8xy + 3xy + 2xy2 – 5x2y (the 2xy2 and the -5x2y can’t be combined, they are not like terms) Solution: 11xy + 2xy2 – 5x2y 13) 12a2b + 5ab2 + 4a2b + 7a2b = 12a2b + 4a2b + 7a2b Solution: 23a2b + 5ab2 + 5ab2 (the 5ab2 can’t be added to the other three terms) 15) 1 𝑥𝑦 2 2 2 3 1 3 + 𝑥𝑦 2 − 𝑥 2 𝑦 Only the first two terms can be combined. I need to get a common denominator of 6, but only for the first two terms. 3 1 2 2 1 = 3 ∗ 2 𝑥𝑦 2 + 2 ∗ 3 𝑥𝑦 2 − 3 𝑥 2 𝑦 3 6 4 6 1 3 = 𝑥𝑦 2 + 𝑥𝑦 2 − 𝑥 2 𝑦 Solution: 7 𝑥𝑦 2 6 1 − 3 𝑥2𝑦 #17 – 26: Add 17) (3x + 5) + (2x – 4) = 3x + 5 + 2x – 4 Solution: 5x + 1 19) (x2 + 6x – 4) + (-5x2 – 3x + 1) = x2 + 6x – 4 + (-5x2) – 3x + 1 = x2 – 5x2 + 6x – 3x – 4 + 1 Solution: -4x2 + 3x – 3 21) (x3 + 5x2 – 3x) + (4x3 – 3x2 + 1) = x3 + 5x2 – 3x + 4x3 – 3x2 + 1 = x3 + 4x3 + 5x2 – 3x2 – 3x + 1 Solution: 5x3 + 2x2 – 3x + 1 23) (8xy + 4x2y + 3) + (17xy – 3x2y – 3) = 8xy + 4x2y + 3 + 17xy – 3x2y – 3 = 8xy + 17xy + 4x2y – 3x2y + 3 – 3 Solution: 25xy + x2y 25) (3ab2 – 4a2b) + (5a2b – 7ab2) = 3ab2 – 4a2b + 5a2b – 7ab2 =3ab2 – 7ab2 – 4a2b + 5a2b Solution: -4ab2 + a2b #27 – 38: Rewrite each problem as a vertical addition problem, then add. 27) (3x2 + 5 – 2x) + (x – 4 +x2) 3𝑥 2 − 2𝑥 + 5 𝑥2 + 𝑥 − 4 ______________ Solution: 4x2 – x + 1 29) (x2 + 6x – 4) + (-5x2 – 3x + 1) 𝑥 2 + 6𝑥 − 4 −5𝑥 2 − 3𝑥 + 1 ______________ Solution: -4x2 + 3x - 3 31) (x2 + 5x3 – 3x) + (4x3 – 3x2 + 5x) 5𝑥 3 + 𝑥 2 − 3𝑥 4𝑥 3 − 3𝑥 2 + 5𝑥 ______________ Solution: 9x3 – 2x2 + 2x 33) (5x – 2) + (x2 + 6x – 4) 5𝑥 − 2 𝑥 2 + 6𝑥 − 4 ______________ Solution: x2 + 11x - 6 35) (y2 + 3y + 6y3) + (2y3 – 3y + 4) 6𝑦 3 + 𝑦 2 + 3𝑦 2𝑦 3 − 3𝑦 + 4 __________________ Solution: 8y3 + y2 + 4 37) (x2 + 3x + 1) + ( 5x – 3) 𝑥 2 + 3𝑥 + 1 5𝑥 − 3 ______________ Solution: x2 + 8x - 2 #39 – 50: Find the opposite of each polynomial 39) 3x – 5 = -(3x – 5) Solution: -3x + 5 41) 4x + 1 = -(4x + 1) Solution: -4x – 1 43) -8x + 9 = -(-8x + 9) Solution: 8x - 9 45) -2x – 5 = -(-2x – 5) Solution 2x + 5 47) -3x2 + 5x – 4 = -(-3x2 + 5x – 4) Solution: 3x2 – 5x + 4 49) 2x3 + 5x – 5 = -(2x3 + 5x – 5) Solution: -2x3 – 5x + 5 51) x4 + 3x3 – 2x2 + 5x – 1 = -( x4 + 3x3 – 2x2 + 5x – 1) Solution: -x4 – 3x3 + 2x2 – 5x + 1 #53-60: Simplify 53) -(3x – 4) Change the signs when you drop the parenthesis Solution -3x + 4 55) -(5x2 – 3x + 2) Solution: -5x2 + 3x – 2 57) -(-2x2 – 3x +1) Solution: 2x2 + 3x - 1 59) -(-x2 + 5x +8) Solution: x2 – 5x – 8 #61 – 70: Subtract 61) (3x + 5) – (2x – 4) = 3x + 5 – 2x + 4 = 3x – 2x + 5 + 4 Solution: x + 9 63) (x2 + 6x – 4) – (-5x2 – 3x + 1) = x2 + 6x – 4 + 5x2 + 3x – 1 = x2 + 5x2 + 6x + 3x – 4 – 4 Solution: 6x2 + 9x - 8 65) (x3 + 5x2 – 3x) – (4x3 – 3x2 + 1) = x3 + 5x2 – 3x – 4x3 + 3x2 – 1 = x3 – 4x3 + 5x2 + 3x2 – 3x – 1 Solution: -3x3 + 8x2 – 3x – 1 67) (8xy + 4x2y + 3) – (17xy – 3x2y – 3) = 8xy + 4x2y + 3 – 17xy + 3x2y + 3 = 4x2y + 3x2y + 8xy – 17xy + 3 + 3 Solution: 7x2y - 9xy + 6 69) (3ab2 – 4a2b) – (5a2b – 7ab2) = 3ab2 – 4a2b – 5a2b + 7ab2 = 3ab2 + 7ab2 – 4a2b – 5a2b Solution: 10ab2 – 9a2b #71 – 82: Rewrite each problem as a vertical subtraction problem, then subtract. 71) (3x2 + 5 – 2x) – (x – 4 +x2) Step 1 3𝑥 2 − 2𝑥 + 5 −(𝑥 2 + 𝑥 − 4) _______________ Step 2 3𝑥 2 − 2𝑥 + 5 −𝑥 2 − 𝑥 + 4 _______________ Solution: 2x2 – 3x + 9 73) (x2 + 6x – 4) – (-5x2 – 3x + 1) Step 1 𝑥 2 + 6𝑥 − 4 −(−5𝑥 2 − 3𝑥 + 1) ___________________ Step 2 𝑥 2 + 6𝑥 − 4 5𝑥 2 + 3𝑥 − 1 __________________ Solution: 6x2 + 9x - 5 75) (x2 + 5x3 – 3x) – (4x3 – 3x2 + 5x) Step 1 5𝑥 3 + 𝑥 2 − 3𝑥 −(4𝑥 3 − 3𝑥 2 + 5𝑥) ______________________ Step 2 5𝑥 3 + 𝑥 2 − 3𝑥 −4𝑥 3 + 3𝑥 2 − 5𝑥 _________________ Solution: x3 + 4x2 – 8x 77) (5x – 2) – (x2 + 6x – 4) Step 1 5𝑥 − 2 −(𝑥 + 6𝑥 − 4) ___________________ 2 Step 2 5𝑥 − 2 −𝑥 − 6𝑥 + 4 ___________________ 2 Solution: -x2 – x + 2 79) (y2 + 3y + 6y3) – (2y3 – 3y + 4) Step 1 6𝑦 3 + 𝑦 2 + 3𝑦 −(2𝑦 3 − 3𝑦 + 4) _______________________ Step 2 6𝑦 3 + 𝑦 2 + 3𝑦 −2𝑦 3 + 3𝑦 − 4) _______________________ Solution: 4y3 + y2 +6y - 4 81) (x2 + 3x + 1) – ( 5x – 3) Step 1 𝑥 2 + 3𝑥 + 1 −( 5𝑥 − 3) _________________ Step 2 𝑥 2 + 3𝑥 + 1 − 5𝑥 + 3 _________________ Solution: x2 – 2x + 4