chapter 8 summary. direct variation if y = kx, then y is said to vary directly as x or be directly...

Chapter 8 Summary

Direct Variation

• If y = kx, then y is said to vary directly as x or be directly proportional to x.– K is the constant of variation– Solve for k first then find the missing value

Inverse Variation

• If y = k/x, then y varies inversely as x.

Joint Variation

• If z = kxy, then z varies jointly as x and y.

Polynomial Division

• Arrange the terms in descending order and don’t forget to insert any “missing terms”

• To divide one polynomial by another, find the quotient and remainder using:– Dividend/Divisor = Quotient + Remainder/Divisor

Synthetic Division

• Can be used instead of long division if the divisor is a first degree polynomial.

Remainder Theorem

• The remainder when P(x) is divided by (x-c) is equal to P(c).– The remainder in synthetic division is the answer

when evaluating a polynomial

Factor Theorem

• The polynomial P(x) has (x – r) as a factor if and only if r is a root of the equation P(x)=0– If a number is factor of the polynomial, the

remainder must be zero when using synthetic division.

Conjugate Root Theorem

• If P(x) has real coefficients and a + bi as a root, then a – bi is also a root.

Depressed Equation

• An equation that results from reducing the number of roots in a given equation by dividing the original equation by one of its factors.

Zeros of Polynomials

• Given p is a polynomial and c is a real number.1. c is a zero of p2. x=c is a solution to p(x)=03. (x-c) is a factor of p(x)4. x=c is an x-intercept of graph p5. C is a zero of p if and only if x – c is a factor!

Difference between root and factor

Root

3

- 5

Factor

X – 3

X + 5

How to solve polynomial equation with degree 3 or higher given a root

1. Use synthetic division with the given root to depress the equation OR Use sum and product to create a polynomial and use long division to depress the equation.

2. Depress the equation until it is an equation you know how to solve.

3. Solve! Don’t forget to write all the roots.

Things to remember when solving

• Use sum and product when solving a polynomial equation with imaginary roots!• Recall: x2 – sum(x) + product

• Graph the polynomial on the calculator and find its zeros to solve!

• The highest degree = number of roots!