01 Polynomials, The building blocks of algebra College Algebra

Numbers Natural / Counting Integers Rational Irrational 1.1 Underlying field of numbers

Real Numbers Irrational

1.2 Indeterminates, variables, parameters Given: ax 2 + bx + c Usual thought: x = variable a, b, & c = constants

mx + c you may recognize and associate this expression with a linear equation The idea (and warning) is to look for definitions Likewise…

Linear equations Most books teach the following: Slope Intercept Form: Standard Form: Point Slope Form: y = mx + b Ax + By = C y – y 1 = m(x - x 1 )

These are the same types of equations c = pn + d pn + c = d Profit = price*quanity - cost

Pythagorean Theorem is a good example also a 2 + b 2 = c 2 What if we are talking about a: Building = B Ladder = L Ground Distance = G L G B B 2 + G 2 = L 2

Variables Used to represent and unknown quantity or a changing value. x y + 2 3x – 2y mx + b

1.3 Basics of Polynomials Parts –Coefficient –Variable –Terms Monomials Polynomial (multiple terms) 3x 2 y + 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 Bases a m * a n 3 2 * 3 4 Add exponents a m+n = 3 6

Rules of Exponents: Exponents Exp raised to an Exp (a m ) n (3 2 ) 4 Multiply exponents a m*n 3 2*4 = 3 8

Rules of Exponents: DIVISION Divide like Bases amam anan Subtract exponents a m-n = 3 2

Rules of Exponents: Qty raised to an Exp (ab) m (3x) 4 Distribute exponents ambmambm 34x434x4 Quantity to an Exponent

Rules of Exponents: Negative Exp Number raised to a neg Exp a -m 3 -2 = the reciprocal 1 amam =

Degrees of Polynomials 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 x 2 y = 3 Degree of xy = 2 Therefore the degree of 3x 2 y + 4xy = 3 3x 2 y + 4xy

Degrees of Polynomials Generally speaking, the degree of 3x 2 y + 4xy = 3 How will this change is y is defined as a constant and x is a variable? 3x 2 y + 4xy

Degrees of Polynomials Generally speaking, the degree of 3x 2 y + 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 3x 2 y + 4xy

1.5 Examples of Polynomial Expressions What is the degree of f(x)? f(x) = x 6 -3x 5 +3x 4 -2x 3 -2x 2 -x+3 What is the degree? 11x 4 y-3x 3 y 2 +7x 2 y 3 -6xy 4 What is the degree if y is a variable? g(x) = 11x 4 y-3x 3 y 3 +7x 2 y 3 -2xy 4

1.5 Examples of “NOW WHAT” happens…Polynomial Expressions f(x) = x 6 -3x 5 +3x 4 -2x 3 -2x 2 -x+3 g(x) = 11x 4 -3x 3 +7x 2 -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) = x 6 -3x 5 +3x 4 -2x 3 -2x 2 -x+3 g(x) = 11x 4 -3x 3 +7x 2 -2x 1.f(x)+g(x) x 6 -3x 5 +3x 4 -2x 3 -2x 2 -x x 4 -3x 3 +7x 2 -2x x 6 -3x 5 +14x 4 -5x 3 +5x 2 -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) = x 6 -3x 5 +3x 4 -2x 3 -2x 2 -x+3 g(x) = 11x 4 -3x 3 +7x 2 -2x 2. f(x)g(x) -- distributive property This could be ugly if one was asked to complete the multiplication (x 6 -3x 5 +3x 4 -2x 3 -2x 2 -x+3 )( 11x 4 -3x 3 +7x 2 -2x)= 11x 10 -3x 9 +7x 8 -2x 7 -33x 9 +9x 8 -21x 7 +6x 6 +33x 8 -9x 7 +21x 6 -6x 5 … what is the degree of the product?

1.5 Examples of “NOW WHAT” happens…Polynomial Expressions f(x) = x 6 -3x 5 +3x 4 -2x 3 -2x 2 -x+3 g(x) = (11x 4 -3x 3 +7x 2 -2x) 3. f(g(x)) (11x 4 -3x 3 +7x 2 -2x) 6 -3(11x 4 -3x 3 +7x 2 -2x) 5 +3(11x 4 -3x 3 +7x 2 -2x) 4 -2(11x 4 -3x 3 +7x 2 -2x) (11x4-3x3+7x2-2x) 2 - (11x4-3x3+7x2-2x) +3 = (11x 4 -3x 3 +7x 2 -2x) 6 - … 11 6 x x x x 6 - … what is the degree?

WebHomework Syntax add subtract multiply divide quantities exponents Be SPECIFIC!!!!! + - * / ( ) ^ Be SPECIFIC!!!!!

WebHomework Syntax 3x 2 y + 4xy 3*x^2*y+4*x*y 4Ab - 5aB 3 4*A*b-5*a*B^3 (Case Sensitive) Quantities ((7+x^2)/(2*z))*y No extra spaces

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