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Chapter 8 Summary

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.

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Chapter 8 Summary

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  1. Chapter 8 Summary

  2. 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

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

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

  5. 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

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

  7. 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

  8. 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.

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

  10. 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.

  11. Zeros of Polynomials • Given p is a polynomial and c is a real number. • c is a zero of p • x=c is a solution to p(x)=0 • (x-c) is a factor of p(x) • x=c is an x-intercept of graph p • C is a zero of p if and only if x – c is a factor!

  12. Difference between root and factor Factor X – 3 X + 5 Root 3 - 5

  13. How to solve polynomial equation with degree 3 or higher given a root • 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. • Depress the equation until it is an equation you know how to solve. • Solve! Don’t forget to write all the roots.

  14. 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!

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