 The degree of a polynomial in one variable is the greatest exponent of its variable.  The coefficient of the variable with the greatest exponent is called the leading coefficient.  If a function is defined by a polynomial in one variable with real coefficients, then it is a polynomial function. If ƒ(x) is a polynomial function the values of x which ƒ(x)=0 are called the zero of the function.

 The solution to a polynomial equation is called the root.  A root or zero may also be an imaginary number.  The imaginary number, i=√-1 and since i=√-1 then i 2 =-1  When imaginary numbers are combined with real numbers, they form complex numbers (in the form of a+bi where a and b are real numbers).  Every polynomial equation with degree greater than zero has at least one root in the set of complex numbers.

 A quadratic equation is a polynomial equation with a degree of 2.  The quadratic formula is: x=[-b±√b 2 -4ac]/2a  You can solve a quadratic equation by using the quadratic formula, completing the square, or factoring.  Solve x 2 -6x-16=0  By Factoring  By Completing the Square  By Quadratic Formula

 It is found under the radical in the Quadratic Equation, aka the radicand DiscriminantNature of Roots/Zeros b 2 -4ac > 02 distinct real roots/zeros b 2 -4ac = 0Exactly one real root/zero (the one real root is actually a double root) b 2 -4ac < 0No real roots/zero (two distinct imaginary roots/zeros)

Word Problem Time!  A rectangular garden is surrounded by a 60- foot long fence. One side of the garden is 6 feet longer than the other. Write an equation that could be used to find s, the shorter side, of the garden?

 For arithmetic you may remember that the dividend equals the product of the divisor and the quotient plus the remainder. For example, 44/7=6 R2, so 44=7(6)+2. This relationship can be applied to polynomials.  Solve 2a+7 / a-2  When a polynomial is divided by one of its binomials factors x-r, the quotient is called a depressed polynomial.  Find the binomial factors of x 3 -7x+6  The Remainder Theorem can be used to determine missing coefficients.  Find the value of k so that the remainder of (x 3 +3x 2 -kx-24) / (x+3) is 0.

 A rational equation has one or more rational expressions.  Solve a + ((a 2 -5)/(a 2 -1))=((a 2 +a+2)/(a+1))  On order to add or subtract fractions with unlike denominators, you must first find a common denominator. Suppose you have a rational expression and you want to know what fractions were added or subtracted to obtain that expression. Finding these fractions is called decomposing the fraction into partial fractions.  Decompose (8y+7)/(y 2 +y-2)  The process used to solve rational equations can be used to solve rational inequalities.  (((x-2)(x-1))/((x-3)(x-2) 2 ))<0.

 Equations in which radical expressions include variables are known as radical equations.  The first step is to isolate the radical on one side, then raise the equation to the proper power to eliminate the radical expression. This process of raising each side of an equation to a power sometimes produces extraneous roots. These are the solutions that do not satisfy the original equation. Therefore, it is important to check all possible solutions.  Solve x=√(x+7)+5  Solve √(x+10)=5-√(3-x)  The same steps are used to solve radical inequalities.  Solve √(4x+5)<10.

FunctionLinearQuadraticCubicQuartic Typical Graph Direction Change 0123

Why do we need to know this?!  Physics  The formula T=2√(l/g) is used to find the period T of a oscillating pendulum. In this formula, l is the length of the pendulum, and g is acceleration due to gravity. Acceleration due to gravity is 9.8 meters per second squared. If a pendulum has an oscillation period of 1.6 seconds, determine the length of the pendulum. (Found on page 271 #59)