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01 Polynomials, The building blocks of algebra College Algebra 1.1 Underlying field of numbers Numbers • Natural / Counting • Integers • Rational • Irrational Real Numbers Irrational 3 5 5 2 3 1.2 Indeterminates, variables, parameters Given: ax2 + bx + c Usual thought: x = variable a, b, & c = constants Likewise… mx + c you may recognize and associate this expression with a linear equation The idea (and warning) is to look for definitions Linear equations Most books teach the following: • Slope Intercept Form: y = mx + b Ax + By = C • Standard Form: y – y1 = m(x - x1) • Point Slope Form: These are the same types of equations • c = pn + d • pn + c = d • Profit = price*quanity - cost Pythagorean Theorem is a good example also a2 + b2 = c2 • • • • What if we are talking about a: Building = B Ladder = L Ground Distance = G B2 + G2 = L2 L B G Variables • Used to represent and unknown quantity or a changing value. x 3x – 2y y+2 mx + b 1.3 Basics of Polynomials • Parts – Coefficient – Variable – Terms • Monomials • Polynomial (multiple terms) 3x2y + 4xy Remember you may have definitions 1.4 Working with Polynomials • To add or subtract one must have like terms. 3xy + 4xy = 7xy 3xy+4x is in simplified form Rules of Exponents: MULTIPLICATION • Multiply like • Add exponents Bases m a * n a m+n a 2 3 4 3 2+4 3 * = 6 3 Rules of Exponents: Exponents • Exp raised to an Exp • Multiply exponents m n (a ) m*n a 2 4 (3 ) 2*4 3 = 8 3 Rules of Exponents: DIVISION • Divide like Bases • Subtract exponents am an m-n a 34 32 4-2 3 = 2 3 Rules of Exponents: Quantity to an Exponent • Qty raised to an Exp • Distribute exponents m (ab) m m a b (3x)4 4 4 3x Rules of Exponents: Negative Exp • Number raised to a neg Exp a-m • = the reciprocal 3-2 12 2 3 1 am 1 =9 Degrees of Polynomials 3x2y + 4xy • Degrees will be dependent on the definition of the variables. • The degree is the highest (combined value) of the exponents of one term. • Degree of x2y = 3 • Degree of xy = 2 Therefore the degree of 3x2y + 4xy = 3 Degrees of Polynomials 3x2y + 4xy • Generally speaking, the degree of 3x2y + 4xy = 3 • How will this change is y is defined as a constant and x is a variable? Degrees of Polynomials 3x2y + 4xy • Generally speaking, the degree of 3x2y + 4xy = 3 • How will this change is y is defined as a constant and x is a variable? • The Degree = 2 because 2 is the highest exponent of the VARIABLE 1.5 Examples of Polynomial Expressions • What is the degree of f(x)? f(x) = x6-3x5+3x4-2x3-2x2-x+3 • What is the degree? 11x4y-3x3y2+7x2y3-6xy4 • What is the degree if y is a variable? g(x) = 11x4y-3x3y3+7x2y3-2xy4 1.5 Examples of “NOW WHAT” happens…Polynomial Expressions f(x) = x6-3x5+3x4-2x3-2x2-x+3 g(x) = 11x4-3x3+7x2-2x 1. f(x)+g(x) 2. f(x)g(x) 3. f(g(x)) 1.5 Examples of “NOW WHAT” happens…Polynomial Expressions f(x) = x6-3x5+3x4-2x3-2x2-x+3 g(x) = 11x4-3x3+7x2-2x 1. f(x)+g(x) x6-3x5+3x4-2x3 -2x2 -x+3 + 11x4-3x3+7x2 -2x x6-3x5+14x4-5x3+5x2-3x+3 Possible questions.. What is the degree? What is the coefficient of the x cubed term? 1.5 Examples of “NOW WHAT” happens…Polynomial Expressions f(x) = x6-3x5+3x4-2x3-2x2-x+3 g(x) = 11x4-3x3+7x2-2x 2. f(x)g(x) -- distributive property This could be ugly if one was asked to complete the multiplication (x6-3x5+3x4-2x3-2x2-x+3)(11x4-3x3+7x2-2x)= 11x10-3x9+7x8 -2x7 -33x9+9x8-21x7+6x6 +33x8-9x7+21x6-6x5 … what is the degree of the product? 1.5 Examples of “NOW WHAT” happens…Polynomial Expressions f(x) = x6-3x5+3x4-2x3-2x2-x+3 g(x) = (11x4-3x3+7x2-2x) 3. f(g(x)) (11x4-3x3+7x2-2x)6-3(11x4-3x3+7x2-2x)5 +3(11x4-3x3+7x2-2x)4-2(11x4-3x3+7x2-2x)32(11x4-3x3+7x2-2x)2-(11x4-3x3+7x2-2x)+3 = (11x4-3x3+7x2-2x)6- … 116x24-36x18+76x12-64x6- … what is the degree? WebHomework Syntax • • • • • • add subtract multiply divide quantities exponents Be SPECIFIC!!!!! • • • • • • + * / ( ) ^ Be SPECIFIC!!!!! WebHomework Syntax • 3x2y + 4xy 3*x^2*y+4*x*y • 4Ab - 5aB3 4*A*b-5*a*B^3 (Case Sensitive) 2 7 x • Quantities y 2z ((7+x^2)/(2*z))*y • No extra spaces Free Mathematics Software • http://math.exeter.edu/rparris/