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Intermediate Algebra 098A

This comprehensive guide covers the fundamentals of exponents and polynomial factoring in Intermediate Algebra. We explore integer exponents, including the product and quotient rules, as well as the significance of coefficients and degrees in monomials and polynomials. Learn how to add, subtract, and evaluate polynomial functions efficiently, along with special multiplication techniques such as FOIL and factoring by grouping. Equipped with practical examples and calculator methods, this resource aims to strengthen your algebraic skills and application in real-life scenarios.

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Intermediate Algebra 098A

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  1. Intermediate Algebra 098A Review of Exponents & Factoring

  2. 1.1 – Integer Exponents • For any real number b and any natural number n, the nth power of b is found by multiplying b as a factor n times.

  3. Exponential Expression – an expression that involves exponents • Base – the number being multiplied • Exponent – the number of factors of the base.

  4. Product Rule

  5. Quotient Rule

  6. Integer Exponent

  7. Zero as an exponent

  8. Calculator Key • Exponent Key

  9. Sample problem

  10. more exponents • Power to a Power

  11. Product to a Power

  12. Polynomials - Review • Addition • and • Subtraction

  13. Objective: • Determine the coefficient and degree of a monomial

  14. Def: Monomial • An expression that is a constant or a product of a constant and variables that are raised to whole –number powers. • Ex: 4x 1.6 2xyz

  15. Definitions: • Coefficient: The numerical factor in a monomial • Degree of a Monomial: The sum of the exponents of all variables in the monomial.

  16. Examples – identify the degree

  17. Def: Polynomial: • A monomial or an expression that can be written as a sum or monomials.

  18. Def: Polynomial in one variable: • A polynomial in which every variable term has the same variable.

  19. Definitions: • Binomial: A polynomial containing two terms. • Trinomial: A polynomial containing three terms.

  20. Degree of a Polynomial • The greatest degree of any of the terms in the polynomial.

  21. Examples:

  22. Objective • Add • and • Subtract • Polynomials

  23. To add or subtract Polynomials • Combine Like Terms • May be done with columns or horizontally • When subtracting- change the sign and add

  24. Evaluate Polynomial Functions • Use functional notation to give a polynomial a name such as p or q and use functional notation such as p(x) • Can use Calculator

  25. Calculator Methods • 1. Plug In • 2. Use [Table] • 3. Use program EVALUATE • 4. Use [STO->] • 5. Use [VARS] [Y=] • 6. Use graph- [CAL][Value]

  26. Objective: • Apply evaluation of polynomials to real-life applications.

  27. Intermediate Algebra 5.4 • Multiplication • and • Special Products

  28. Objective • Multiply • a • polynomial • by a • monomial

  29. Procedure: Multiply a polynomial by a monomial • Use the distributive property to multiply each term in the polynomial by the monomial. • Helpful to multiply the coefficients first, then the variables in alphabetical order.

  30. Law of Exponents

  31. Objectives: • Multiply Polynomials • Multiply Binomials. • Multiply Special Products.

  32. Procedure: Multiplying Polynomials • 1. Multiply every term in the first polynomial by every term in the second polynomial. • 2. Combine like terms. • 3. Can be done horizontally or vertically.

  33. Multiplying Binomials • FOIL • First • Outer • Inner • Last

  34. Product of the sum and difference of the same two termsAlso called multiplying conjugates

  35. Squaring a Binomial

  36. Objective: • Simplify Expressions • Use techniques as part of a larger simplification problem.

  37. Albert Einstein-Physicist • “In the middle of difficulty lies opportunity.”

  38. Intermediate Algebra –098A • Common Factors • and • Grouping

  39. Def: Factored Form • A number or expression written as a product of factors.

  40. Greatest Common Factor (GCF) • Of two numbers a and b is the largest integer that is a factor of both a and b.

  41. Calculator and gcd • [MATH][NUM]gcd( • Can do two numbers – input with commas and ). • Example: gcd(36,48)=12

  42. Greatest Common Factor (GCF) of a set of terms • Always do this FIRST!

  43. Procedure: Determine greatest common factor GCF of 2 or more monomials • 1. Determine GCF of numerical coefficients. • 2. Determine the smallest exponent of each exponential factor whose base is common to the monomials. Write base with that exponent. • 3. Product of 1 and 2 is GCF

  44. Factoring Common Factor • 1. Find the GCF of the terms • 2. Factor each term with the GCF as one factor. • 3. Apply distributive property to factor the polynomial

  45. Example of Common Factor

  46. Factoring when first terms is negative • Prefer the first term inside parentheses to be positive. Factor out the negative of the GCF.

  47. Factoring when GCF is a polynomial

  48. Factoring by Grouping – 4 terms • 1. Check for a common factor • 2. Group the terms so each group has a common factor. • 3. Factor out the GCF in each group. • 4. Factor out the common binomial factor – if none , rearrange polynomial • 5. Check

  49. Example – factor by grouping

  50. Ralph Waldo Emerson – U.S. essayist, poet, philosopher • “We live in succession , in division, in parts, in particles.”

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