Topic 6.2       (Polynomial Operations I)

Degree - The degree of a monomial is the sum of the exponents of the variables

                    in that term.

Example:            x³y⁴z           Degree is 8.

The degree of a polynomial is the monomial with the highest degree.

Example:             x² + xy³ - 2x⁴y⁵ + 5

Degree:                2      4          9        0     Degree for the 4 termed expression is 9.

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Evaluating polynomials

Substitute the value (number) for the variable and use the order of operations.

     6x² + 4xy - 2y²    Let x = 2.  y = 3.       Substitute the number for the variable.

    6(2)² + 4(2)(3) - 2(3)²                           Simplify

         6(4) + 4(6) - 2(9)

               24 + 24 - 18

                  48 - 18     =     30      Ans.

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When adding or subtracting polynomials follow the steps given in the PAN.

When multiplying a monomial by a polynomial, multiply the monomial by each

   term in the polynomial.

When dividing a monomial by a polynomial, divide each term of the polynomial by

   the monomial.

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Sample Problems

Find:     (-3w - 12w³ + 2) + (12w - 2w³ + 4w⁵ - 3)  Drop parenthesis and combine like

                                                                                         terms.

                -3w - 12w³ + 2 + 12w - 2w³ + 4w⁵ - 3      Put answer in decreasing order

                                                                                         (largest exponent first).

                -3w + 12w - 12w³ - 2w³ + 2 - 3 + 4w⁵      

                               4w⁵ - 14w³ + 9w - 1                      Answer

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Find:        (2y³ + 6y² + 2) - (5y + y³ + 4y⁷ - 3)            Drop parenthesis, change the sign

                                                                                         of each term of the polynomial that

                    2y³ + 6y² + 2 - 5y - y³ - 4y⁷ + 3                that is preceded by a minus.

                      -4y⁷ + y³ + 6y² - 5y + 5                            Answer

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Find:          [-6t³u²v¹¹][(1/2) tu²v⁴]                          Multiply the numerical coefficients

                                                                                         and the variables.

                                 -3t⁴u⁴v¹⁵                                      Answer                                      

Remember, when multiplying variables, add the exponents of the same variable.

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Find:              t³uv⁴(2tu - 3uv + 4tv + 5)                   Multiply the monomial by each

                                                                                          term of the polynomial.

                 2t⁴u²v⁴ - 3t³u²v⁵ + 4t⁴uv⁵ + 5t³uv⁴        Answer

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Find:    (20t⁵u¹¹ + 5t³u⁵ + 30tu⁶v⁵) /  10t⁴u⁵         Divide each term of the polynomial

                                                                                          by the monomial.

   (20t⁵u¹¹) / 10t⁴u⁵  +  (5t³u⁵) / 10t⁴u⁵  +   (30tu⁶v⁵ )/ 10t⁴u⁵       Reduce each

                                                                                                                         fraction.

               2tu⁶               +      1/2t                    +           3uv⁵ / t³                Answer