To Add polynomials, just combine like terms, by adding their coefficients. 1) (2x2 + 5x – 2) + (3x2 – x + 5) 2) Add 5z2 – 10z – 12 3z2 + 7z – 9 To Subtract, you must distribute the negative and then add. 3) (14y3 – 6y2 + 2y – 5) – (2y3 – 7y2 – 4y + 6) 4) Subtract 3mn + 2m – 4n -mn + 4m + n When multiplying all monomials, just use the rules of exponents. When multiplying a monomial by a polynomial (other than another monomial), distribute. 2) When multiplying two binomials, distribute both terms, the acronym FOIL may be used. 3) Another Method for multiplying is to work vertically. 4) Multiplying a sum and a difference of two binomials: 1) 2)
To Square a Binomial: 1.) Square the first term 2.) Double the Product of 1st term and 2nd term 3.) Square the second term 3) 4) Caution: *Using order of operations on polynomials 5) 6) 7) MA 035 Review: Integer Exponents Goals: Use and understand product and power rules for exponents. Exponential Expression: an is an exponential expression, a is the base, n is the exponent and this is shorthand for a times itself n times. Practice: a) 25 b) (-3)3 c) (-4)2 d) -42
2) 3) 4) 5) Complex Numbers: If a and b are real numbers, then any number of the form: a +bi is called a complex number. a is the real part and b is the imaginary part. Since i is a square root, all operations follow the same rules used for simplifying radical expressions. Add: 1) (4 + 5i) + (7 + 3i) 2) (-2 + i) + (5 – 7i) Subtract: 3) (6 + 5i) – (8 + 11i) 4) [(4 – 2i) – (3 + 5i)] – (2 + i) Multiply: 4) 4i(2 + 3i) 5) (2 + i)(3 + 2i) 6) (4 + 6i)(3 – 2i) 7) (3 + 2i)(3 – 2i) Divide: For quotients use the ‘complex conjugate’ to clear i from the denominator and write your answer in the form a + bi 8) 9)
10) 11) Simplifying powers of i: Powers of i are cyclic. Any power of i can be simplified to i,-1, -i, 1. If you know these first 4 powers of i, you can figure out any power of i. i1 = i i2 = -1 i3 = i2i1=-i i4 = i2i2= 1 Watch what happens as we continue: i5 = i4i = 1i = i i6 =i4i2 =1(-1) = -1 i7 = i4i3=-i i8 = i4i4= 1 The powers of i cycle every fourth power. We can use this to find any power of i. i14= i12i2 = (i4)3i2=1(-1) = -1 i33 = i32i = 1i = i i12 i31 i-2 i50
Negative 2 Imaginary Use the discriminant to describe how many and what types of solutions (roots) there will be? 8) 9) 10) MA 035 Notes 10.4: Quadratic Formula and Applications Goals: Solve equations that must be put in quadratic form by using quadratic formula. Solve application problems. Clear the denominators by using the LCD, then apply quadratic formula: 1) 2) 3) Remember that when dealing with rational equations, you must check your solution against the domain. 4) 5)
3) x 4 2 5 6 y 3 1 1 7 B) To determine if an equation is a function, First write it in the form y = a quantity with the variable x. Consider what happens for each input x, is there only 1 output y ? Are these equations functions? 1) 2) 3) 4) 5) 6) C If a graph passes the ‘vertical line test’, it is a function. Vertical Line Test: If every vertical line intersects the graph of a relation only once, then the relation is a function. Are these graphs functions? 1) 2) 3) 4) 5)
Now we will subtract. -4p2 – 5p + 9 And continue the division process. p goes into -4p2 , -4p times -4p2 – 5p + 9 Now subtract, remember to change the signs. -4p2 -8p 3p + 9 continues ---> p goes into 3p +9 3 times -4p2 – 5p + 9 -4p2 -8p Subtracting leaves 3, which p does not divide into 3p + 9 So we will write 3 as a remainder 3p + 6 3 Answer: Practice: 1) 2)