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Polynomial Division

Polynomial Division. Long Division. A. 2 x 3 y – 3 x 5 y 2 B. 1 + 2 x 3 y – 3 x 5 y 2 C. 6 x 4 y 2 + 9 x 7 y 3 – 6 x 9 y 4 D. 1 + 2 x 7 y 3 – 3 x 9 y 4. Division Algorithm. f(x) = d(x) * q(x) + r(x) Dividend=divisor*quotient+remainder

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Polynomial Division

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  1. Polynomial Division

  2. Long Division

  3. A. 2x3y – 3x5y2 B. 1 + 2x3y – 3x5y2 C. 6x4y2 + 9x7y3 – 6x9y4 D. 1 + 2x7y3 – 3x9y4

  4. Division Algorithm • f(x) = d(x) * q(x) + r(x) • Dividend=divisor*quotient+remainder • If r(x) = 0, the d(x) divides evenly into f(x)

  5. x(x – 5) = x2 – 5x –2x – (5x) = 3x 3(x – 5) = 3x – 15 Division Algorithm Use long division to find (x2 – 2x – 15) ÷ (x – 5). Answer: The quotient is x + 3. The remainder is 0.

  6. Polynomial Division Divide by x – 2

  7. Polynomial Division

  8. Aa + 3 B C D Quotient with Remainder Which expression is equal to (a2 – 5a + 3)(2 – a)–1?

  9. Examples by

  10. 1 – 4 6 – 4 1 Synthetic Division Use synthetic division to find (x3 – 4x2 + 6x – 4) ÷ (x – 2). Step 1Write the terms of the dividend so that the degrees of the terms are in descending order. Then write just the coefficients as shown. x3 – 4x2 + 6x – 4  1 – 4 6 – 4 Step 2Write the constant r of the divisor x – r to the left. In this case, r = 2. Bring the first coefficient, 1, down as shown.

  11. 1 –4 6 – 4 2 1 –2 1 –4 6 – 4 2 –4 2 1 –2 Synthetic Division Step 3 Multiply the first coefficient by r : 1 ● 2 = 2. Write the product under the second coefficient. Then add the product and the second coefficient. Step 4Multiply the sum, –2, by r : 2(–2) = –4. Write the product under the next coefficient and add: 6 + (–4) = 2.

  12. 1 –4 6 – 4 4 2 –4 1 2 0 –2 Synthetic Division Step 5 Multiply the sum, 2, by r : 2(2) = 4. Write the product under the next coefficient and add: –4 + 4 = 0. The remainder is 0. The numbers along the bottom are the coefficients of the quotient. Start with the power of x that is one less than the degree of the dividend. Answer: The quotient is x2 – 2x + 2.

  13. Example Use synthetic division to find (x2 + 8x + 7) ÷ (x + 1).

  14. Divide by x + 3

  15. Now it’s your turn! Divide

  16. Divide numerator and denominator by 2. Simplify the numerator and denominator. Divisor with First Coefficient Other than 1 Use synthetic division to find (4y4 – 5y2 + 2y + 4) ÷ (2y – 1). Use division to rewrite the divisor so it has a first coefficient of 1.

  17. Finish the example

  18. Remainder Theorem • If a polynomial f(x) is divided by x – k, then the remainder is r = f(k) Find the remainder of divided by x + 2

  19. Factor Theorem • A polynomial f(x) has a factor (x – k) if and only if f(k) = 0. Completely Factor if (x + 4) is a factor.

  20. Example • Completely Factor

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