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1 Warm-up Determine if the following are polynomial functions in one variable. If yes, find the LC and degree Given the following polynomial function, find the following information to help graph it! Degree? LC? End Behavior? Y-int, and Factors with Multiplicity? Graph it!

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2 Warm-up NO Yes, LC: -4; Degree: 5 Degree: 4; LC 1 Y-Int: (0, 0) Factors: x 2 (x-3)(x+2) x = 0 (m=2, even) x = 3 (m = 1; odd) x = -2 (m = 1; odd) End Behavior Left: Up Right: Up

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Using Synthetic Division to find Zeros Section 2-3

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4 Objectives I can use synthetic division to find factors of a polynomial I can use synthetic division to find zeros of a polynomial

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5 Dividing Numbers When you divide a number by another number and there is NO REMAINDER: Then the DIVISOR is a factor!! Also the QUOTIENT becomes another factor!!! Dividend Divisor Quotient

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6 Find: (6x 3 - 19x 2 + x + 6) (x-3) 6x 3 – 19x 2 + 1x + 6 6 -19 1 6 3 6 18 -3 -2 -6 0 6x 2 – 1x – 2 (No remainder) That means (x – 3) is a factor and (6x 2 – x – 2) is also a factor That means (3, 0) is a zero.

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7 Find: (4x 4 - 5x 2 + 2x + 4) (x+1) 4x 4 + 0x 3 – 5x 2 + 2x + 4 4 0 -5 2 4 4 -4 4 1 3 -3 1 That means (x + 1) is NOT a factor

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8 Finding Additional Factors or Zeros Sometimes you will know one factor or zero, but need to find the remaining factors or zeros Then using synthetic division we would divide by the known factor or zero and the quotient will be a new factor.

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9 Given: x 3 - x 2 - 5x - 3; (x + 1) 1x 3 – 1x 2 – 5x – 3 1 -1 -5 -3 1 -2 2 -3 3 0 1x 2 – 2x - 3 ( )( ) (x – 3)(x + 1) (3, 0) (-1, 0)

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10 Given: x 3 + 5x 2 - 12x - 36; (3, 0) 1x 3 + 5x 2 – 12x – 36 3 1 5 -12 -36 1 3 8 24 12 36 0 1x 2 + 8x - 12 ( )( ) (x + 6)(x + 2) (-6, 0) (-2, 0)

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11 Factor Theorem Factor Theorem: A polynomial f(x) has a factor (x – k) if and only if f(k) = 0. Example: Show that (x + 2) and (x – 1) are factors of f(x) = 2x 3 + x 2 – 5x + 2. 6 2 1 – 5 2 – 2 2 – 4 – 31 – 2 0 The remainders of 0 indicate that (x + 2) and (x – 1) are factors. – 1 2 – 3 1 1 2 2 – 10 The complete factorization of f is (x + 2)(x – 1)(2x – 1).

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Read the Question Find the remaining factors Find all the factors Find the remaining zeros Find all the zeros 12

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13 Homework WS 4-2

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