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A polynomial function is a function of the form

OFFICIAL POLYNOMIAL JARGON. a n. n. n. n – 1. a 0. a n  0. leading coefficient. a n. constant term. degree. a 0. n. descending order of exponents from left to right. A polynomial function is a function of the form. f ( x ) = a n x n + a n – 1 x n – 1 +· · ·+ a 1 x + a 0.

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A polynomial function is a function of the form

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  1. OFFICIAL POLYNOMIAL JARGON an n n n– 1 a0 an 0 leading coefficient an constant term degree a0 n descending order of exponents from left to right. A polynomial function is a function of the form f(x) = an xn+ an– 1xn– 1+· · ·+ a1x + a0 Where an 0 and the exponents are all whole numbers. For this polynomial function, an is the leading coefficient, a0 is the constant term, and nis the degree. A polynomial function is in standard form if its terms are written in descending order of exponents from left to right.

  2. 3 6 1 8 -1 8 0 4 - 3 1 0 - 9 1 Solution is: 2613 Let’s remember how to do long division. 2 6 1 3 7 8 4 0 Remainder = 1

  3. x (x + 4) x3 + 6x2 + 2x + 10 Use the 1st term to determine what to multiply by. + 2x 2x2 - 6x + 10 34 This is the remainder! Solution is: x2 + 2x – 6 + So now, we’re going to use the same process to divide polynomials. x2 + 2x - 6 + 4x2 x3 x times what equals x3? 2x2 + 8x x times what equals 2x2? - 24 - 6x x times what equals –6x?

  4. (x + 6) 2x3 + 15x2 + 20x + 10 + 20x 3x2 2x + 10 -2 This is the remainder! Solution is: 2x2 + 3x + 2 + You try . . . 2x2 + 3x + 2 + 12x2 2x3 x times what equals 2x3? + 18x 3x2 x times what equals 3x2? 2x + 12 x times what equals 2x?

  5. Use synthetic division to divide 2x4 + -8x2 + 5x- 7 by (x – 3). This is in the format (x – k). Using Synthetic Division Another way to divide a polynomial is to use synthetic division. What is k if you’re dividing by (x + 7) ? -7 So, k = 3

  6. Using Synthetic Division Polynomial in standard form 3 Coefficients k-value Solution! SOLUTION 2x4 + 0x3 + (–8x2) + 5x + (–7) Polynomial in standard form 2 0 –8 5 –7 3• Coefficients 6 18 30 105 35 10 98 2 6 X3-term X2-term Remainder Constant x-term + 2 x3 + 6 x2 + 35 + 10 x

  7. Polynomial in standard form -2 Coefficients k-value Solution! You Try . . . Divide 3x4 + 4x3 – 8x – 20 by (x + 2) 3x4 + 4x3 + 0x2 + -8x + -20 Polynomial in standard form 3 4 0 -8 -20 -2• Coefficients -6 4 -8 32 -16 4 12 3 -2 X3-term X2-term X-term Remainder Constant - 16 + 3 x3 + 4 x - 2 x2

  8. 1 7 -8 1 1 8 8 0 1 Remainder x-term constant x + 8 A Few Notes --The divisor must be of the form (x – k). --If it is not, then you must use long division (i.e. 2x – 7). --If the remainder is 0, then the answer is a factor of the original polynomial! Divide x2 + 7x – 8 by (x – 1) (x – 1)(x + 8) = x2 + 7x – 8!! --Something really cool  f(k) = the remainder !!

  9. Try some more. Divide. x2 - 2x - 8 (x - 4) Divide. 3x3 - 2 x2 + 5x - 1 (x + 2) Divide. 4x3 - 7x + 8 (2x - 1)

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