Section 2.1-2.2 September 6, 2017 1 Polynomials Definition. A polynomial is an expression of the form a n x n + a n 1 x n 1 + + a 1 x + a 0 where each a 0, a 1,, a n are real numbers, a n 0, and n is a positive integer. Remarks: 1. n is called the degree of the polynomial. 2. The a i are called the coefficients. 3. a n is called the leading coefficient. 4. a 0 is the constant term 5. If n = 1, the polynomial is linear. 6. If n = 2, the polynomial is called a quadratic 7. If n = 3, 4, 5,..., the polynomial is called a cubic, quartic, quintic, etc. Example. The height, s (in feet), of a ball thrown in the air at time t = 0 seconds is given by the polynomial equation s = 16t 2 + 80t + 5 This is a polynomial with degree: leading coefficient: constant: Note. Polynomial expressions do not contain negative exponents or radicals/fractional exponents. Only positive integer exponents of a single variable and a constant. Example. Are these polynomials? 1. 3x 7 4x + 5 x 2. 3z 2 + 5z z + 8 3. x 3 + 7y 2 3 y 3 + 10 1
Sums and Differences of Polynomials: Addition and subtraction of polynomials is done by combining like terms, that is, terms which have the same variable and exponent. Example. Simplify the expressions 1. (7x 3 2x + 3) + (x 4 + 5x + 9) 2. (13x 2 x) (3x 2 5x + 7) 3. (13x 100 + 500x 53 + 3) (10x 100 400x 53 + 50) Products of Polynomials: Multiplication of polynomials is based on the Distributive Property of the Real Numbers. Each term in the first polynomial is multiplied by the entire second polynomial. Example. Expand (multiply out) the following expressions. 1. (x 7)(5x + 8) 2. (3x 7)(5x 3 2x 2 4) 3. (a + b)(a b) 2
Formulas: These are common products of polynomials that are useful to multiply or factor polynomials. Difference of Squares: a 2 b 2 = (a + b)(a b) Square of a Binomial: (a + b) 2 = a 2 + 2ab + b 2 (a b) 2 = a 2 2ab + b 2 Sum of Cubes: a 3 + b 3 = (a + b)(a 2 ab + b 2 ) Difference of Cubes: a 3 b 3 = (a b)(a 2 + ab + b 2 ) Cube of a Binomial: (a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3 (a b) 3 = a 3 3a 2 b + 3ab 2 b 3 Note. There is NO special product (in the real numbers) for the sum of squares a 2 + b 2. Example. Expand the expressions 1. (x 4) 3 2. [(x 3y)(x + 3y)] 2 Quotients of Polynomials: Division of polynomials is done using standard long division techniques: Divide, Multiply, Subtract, Bring Down, Repeat Example. Use long division to find 2372 13 3
Example. Use long division to divide the polynomials: 1. (2x 2 4x 16) (x 4) 2. (x 6 + 3x 5 + 2x 3 + x 4) (x 3 x) 3. (2x 7 + 3x 5 + 4x 4 + 2x 2 + 3) (x 4 + 2x 2 5) 4
2 Factoring Common Factors: A common factor is a factor of every term of an expression. Common factors can be pulled out of an expression using the distributive property in reverse. Example. Factor out any common factors: 1. x 7 + 4x 3 5x 2 = 2. 6x 3 y 5 z 3 8x 2 y 3 12xy 6 z 2 = 3. 3x 3 x + 5 + x x + 5 Factoring by Grouping: Factoring by grouping is useful when you have more than three terms in the polynomial. Factoring by Grouping separates the expression into groups and then uses common factors. Example. Factor the following polynomials completely 1. 3x 3 5x 2 + 12x 20 2. 16x 5 + 24x 3 54x 2 81 Note. An expression is factored completely if it only has three kinds of terms left: 1. single factors (terms with no +): 7, x 7, y 8, x, etc. 2. linear polynomials ax + b where a and b are relatively prime (have no common factors) 3. irreducible quadratics: ax 2 + bx + c (You must check that it actually is irreducible and cannot be factored) 5
Factoring Using Special Products We should check if the expression has the form of a special product. Difference of Squares: a 2 b 2 = (a + b)(a b) Square of a Binomial: (a + b) 2 = a 2 + 2ab + b 2 (a b) 2 = a 2 2ab + b 2 Sum of Cubes: a 3 + b 3 = (a + b)(a 2 ab + b 2 ) Difference of Cubes: a 3 b 3 = (a b)(a 2 + ab + b 2 ) Cube of a Binomial: (a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3 (a b) 3 = a 3 3a 2 b + 3ab 2 b 3 Note. There is NO special product (in the real numbers) for the sum of squares a 2 + b 2. Example. Factor the expressions completely. 1. 5x 2 80 2. 16x 4 54x 3. 3x 4 + 75x 2 4. x 3 + 9x 2 + 27x + 27 6
Factoring Quadratics ax 2 + bx + c 1 We want factors m and n with mn = ac and m + n = b. So we start by finding all the factors of ac and trying to find a pair that adds to b. 2 Split the middle term bx into two terms mx and nx and factor by grouping. Example. Factor completely. 1. x 2 x + 12 2. 10x 2 + 31x + 15 3. 28x 2 + 17x 3 Factoring Out a Common Base to the Lowest Power: Example. Factor the following expressions completely. 1. 5x 3 y 2 3 + 30x 2 y 7 3 2. 6x 11 5 + 2x 6 5 20x 1 5 7
In general, you will need to use all the above techniques to completely factor expressions, and you will not be told which ones to use. Example. Factor completely. 1. 12x 2 + 52x 40 2. 9x 4 y 2 81x 3 y 2 3. 5x 5 45x 3 5x 2 + 45 4. 2x 7 128x 4 5. 6x 8 5 y 2 24x 2 5 8
3 Rational Expressions Definition. A rational expression is the quotient p q of two polynomials, p and q. Polynomials are also rational expressions with denominator 1. Example. The following are rational expressions: Example. These are not rational expressions: 5x + 3 x 2 7, 3x 3 4x + 10 x 6 5x 4 + 3x, 2x2 + 3 3x 2 7 x 2 4, sin x 2 3x 5 2x 2 + 3 Domain: The domain of a rational expression is all the values of the variable where the expression is defined, which is all the values where the denominator is not zero. Example. Find the domain of the following rational expressions: 1. x 2 7x + 8 3x 11 2. 3x 4 x 2 + 36 Simplifying Rational Expressions: To simplify a rational expression, factor the polynomials in the numerator and denominator and cancel the common factors. Make sure to restrict the domain of the expression as necessary. Examples: 1. 5x 5 40x 2 x 4 4x 2 2. 3x 4 + 7x 3 20x 2 5x 4 + 5x 3 60x 2 9
Operations with Rational Expressions Note. Rational expressions behave like fractions (just the numerator and denominator are polynomials instead of numbers). 1. Multiplication of Rational Expressions Rational expressions multiply just like fractions: multiply the numerators and multiply the denominators. a b c d = ac bd Then simplify the rational expression. 2. Division of Rational Expressions Rationals expressions divide like fractions: dividing by a rational expression is the same as multiplying by its reciprocal (the reciprocal switches the numerator and denominator). a b c d = a b d c Example. Perform the operation and simplify. List all restrictions on x. 1. 2x 6 x 1 2x + 4 x 3 2. 2x 2 + x 6 x 2 + 4x 5 x3 3x 2 + 2x 4x 2 x + 5 6x x 3 8 3. x 3 27 x 4 5x 3 x2 + x 12 x 2 x4 + 4x 3 25 x 2 9 4. 4x + 20 36 x 2 x 3 6x 2 2x 2 4x 16 3x2 + 15x 2x 2 3x 10
3. Addition and Subtraction of Rational Expressions Rational expressions add and subtract just like fractions: you need to find a least common denominator (LCD) and convert each fraction to a fraction whose denominator is the LCD. To find the LCD of several fractions: (1). Factor all the denominators (2). The LCD is the product of all the prime factors of the denominators, and the exponent of each prime factor is the highest exponent out of all the denominators. Example. Perform the operation and simplify. List any restrictions on the variable. 1. 2x + 4 x 3 + x 5 x + 1 2. y 4 y 2 2y 3y2 + 7 y 2 3y + 2 Compound Fractions: A compound fractions (or complex fraction) is an expression which contains nested fractions, usually one main fraction which will have one or more fractions in its numerator and/or denominator. To simplify a compound fraction: 1. Simplify the numerator and denominator of the main fraction. 2. Perform the division by multiplying by the reciprocal of the denominator. Example. Simplify the expression and list any restrictions on the variables. 1. 6 y 5 2y+1 6 y + 4 11
2. 5(x + h) 3 5x 3 h Example. For the function f(x) = 3x f(x+h) f(x) x+4, find and simplify h. 12