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Polynomials

Polynomials. Polynomial functions: f(x) = 4x 3  + 8x 2  + 2x + 3, g(x) = 2.5x 5  + 5.2x 2  + 7, h(x) = 3x 2 Polynomial functions are functions that have this form: f(x) = a n x n  + a n-1 x n-1  + ... + a 1 x + a 0, a n

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Polynomials

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  1. Polynomials

  2. Polynomial functions: • f(x) = 4x3 + 8x2 + 2x + 3, g(x) = 2.5x5 + 5.2x2 + 7, h(x) = 3x2 • Polynomial functions are functions that have this form: • f(x) = anxn + an-1xn-1 + ... + a1x + a0, an • The value of n must be an nonnegative integer (0, 1, 2, ….) • The coefficients of each term, an, an-1, ..., a1, a0 are real numbers. • The degree of the polynomial function is the highest value forn.

  3. Determine the degree of the following polynomials: (a) (b) (c) Determine whether the functions below are polynomials: (a) (b) (c) (d)

  4. Operations with polynomials: • Addition • Example: 1.(2x + 5y) + (3x – 2y) =  2x + 5y + 3x – 2y=  2x + 3x + 5y – 2y=  5x + 3y 2. (3x3 + 3x2 – 4x + 5) + (x3 – 2x2 + x – 4)   =  3x3 + 3x2 – 4x + 5 + x3 – 2x2 + x – 4   =  3x3+ x3 + 3x2 – 2x2 – 4x + x + 5 – 4   =  4x3 + 1x2 – 3x + 1 ...or vertically:   

  5. Subtraction Simplify(x3 + 3x2 + 5x – 4) – (3x3 – 8x2 – 5x + 6) (x3 + 3x2 + 5x – 4) – (3x3 – 8x2 – 5x + 6)   =  (x3 + 3x2 + 5x – 4) – 1(3x3 – 8x2 – 5x + 6)   =  (x3 + 3x2 + 5x – 4) – 1(3x3) – 1 (–8x2) – 1(–5x) – 1(6)   =  x3 + 3x2 + 5x – 4 – 3x3 + 8x2 + 5x – 6   =  x3 – 3x3 + 3x2 + 8x2 + 5x + 5x – 4 – 6   =  –2x3 + 11x2 + 10x –10

  6. Multiplication (4x2– 4x– 7)(x + 3) = (4x2– 4x– 7)(x) + (4x2– 4x– 7)(3) = 4x2(x) – 4x(x) – 7(x) + 4x2(3) – 4x(3) – 7(3) = 4x3– 4x2– 7x + 12x2– 12x– 21 = 4x3– 4x2 + 12x2– 7x– 12x– 21 = 4x3 + 8x2– 19x– 21 or you may do it vertically:

  7. Division Case 1: To divide a polynomial by a monomial, we can divide each term by the monomial. Example: Simplify

  8. Division Case 2: To divide a polynomial by a more complicated polynomial, factor up to simplify if possible or if not, using the long division method Example: 1. Simplify

  9. Division • Case 3 : When a polynomial a(x), is divided by a non-constant divisor, b(x), the quotient q(x) and the remainder r(x) are defined by the identitiy • polynomial divisor quotient remainder • Degree of a(x) = degree of b(x) + degree of q(x) • Degree of r(x) must be less than the degree of b(x) Eg. Divide 3x3 – 5x2 + 10x – 3  by  3x + 1 3x3 – 5x2 + 10x – 3 =(3x+1)(x2 – 2x+4) – 7 degree 3 degree 1 degree 2 degree 0

  10. Long division method

  11. Please note that • the remainder is at least one degree less than the divisor • the sum of the degree of quotient and divisor is the degree of the polynomial quotient divisor remainder 3x3 – 5x2 + 10x – 3 =(3x+1)(x2 – 2x+4) – 7

  12. Synthetic Division • Most commonly used for linear divisor in the form of Eg. Divide by

  13. Activity • Work in groups of 3. Solve the following problems. • Time: 10 minutes • Simplify • Find • Use long division to find (i) (ii)

  14. Zeros and Factors of Polynomials Zero (or root) of a polynomial -solution to the equation. Find the zeros of the polynomials below: (a) (b) (c)

  15. Fundamental Theorem of Algebra If  is a polynomial of degree n then  will have exactly n zeroes, some of which may repeat. -a polynomial of degree 3 will have ____ zeroes/ roots -a polynomial of degree 6 will have ____ zeroes/ roots Eg.

  16. The Remainder Theorem (for linear divisor only) When a polynomial is divided by , the remainder is the constant . When a polynomial is divided by , the remainder is the constant .

  17. Example: Find the remainder of p(x) = x3 – 7x – 6 when it is divided by x- 4. Solution: The remainder is p(4) = 43 – 7(4) – 6 = 30.

  18. The Factor Theorem (for linear divisor only) For the polynomial , If  is a factor of   , then . If ,  then   will be a factor of . For the polynomial , If  is a factor of   , then . If ,  then   will be a factor of .

  19. The Factor Theorem Example: Find the value of a for which is a factor of ? Solution: Using the factor theorem,

  20. The Factor Theorem Find the factors of the following cubic polynomial. Solution: factors of 6: Working through these factors, eventually we find that

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