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Polynomial Expressions

Polynomial Expressions. Section P.3. Definition of Polynomial. An algebraic expression of the form Where all coefficients a i are real numbers, The degree n is a nonnegative integer. Skills. Which of the following are polynomial expressions?. Operations on Polynomials.

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Polynomial Expressions

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  1. Polynomial Expressions • Section P.3

  2. Definition of Polynomial • An algebraic expression of the form • Where all coefficients ai are real numbers, • The degree n is a nonnegative integer

  3. Skills • Which of the following are polynomial expressions?

  4. Operations on Polynomials • We can combine polynomials in a number of ways • Addition of polynomial expressions • Subtraction of polynomial expressions • Multiplication of polynomial expressions

  5. Addition of Polynomials • We can add polynomials by using the commutative and associative properties of real numbers to group like terms. Add the following polynomial expressions.

  6. Multiplication of Polynomials • Multiplying two polynomials requires the distributive property. The rule for multiplication can be interpreted geometrically.

  7. Geometric Representation • (x + 1) (3x + 2) 2 3x 2x x 3x2 3x 2 1

  8. Multiplication Rule • Multiplying polynomials can be done by applying the distributive property • Try this:

  9. Multiplication Rule • FOIL Mnemonic – a memory aid for multiplying two binomials. Outer First Inner Last

  10. Factoring Polynomials • Relate factoring polynomials to factoring integers • Factoring is looking for patterns and testing cases

  11. Factoring Polynomials • Factor 6x2 + 7x – 20 over the integers • Looking for two binomial factors • 6x2 + 7x – 20 = ( )( ) • Examine the signs. How can we tell what the signs will be? • Pattern: Second sign is – so the signs are opposite. Why? 6x2 + 7x – 20 = ( + )( - )

  12. Factoring Polynomials • What are the possible factors of the first term and last term? • Solution: 6x2 factors are x·6x, 2x·3x • Solution: 20 factors are 1·20, 2·10, 4 ·5 • Guess and check – need difference of 7x for middle term • 6x2 + 7x – 20 = (2x + 5)(3x - 4)

  13. Geometric Representation • If a polynomial factors over the integers, regions representing x2, x and units can be arranged in a rectangle. x x x 1 1 1 1 x 1

  14. Special Factor Rule • Factor the following to determine some special factoring rule

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