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# Theorems on polynomial functions

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Remainder theorem, Factor theorem, Rational Zeroes theorem

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### Theorems on polynomial functions

1. 1. Theorems on Polynomial Functions PSHS Main Campus July 13, 2012PSHS Main Campus () Theorems on Polynomial Functions July 13, 2012 1/7
2. 2. Remainder TheoremRemainder TheoremWhen a polynomial P (x) is divided by (x − a), the remainder is P (a).Examples 1 Find f (3) if f (x) = 2x3 − 5x2 − 8x + 17. 2 Find g(−5) if g(x) = x4 − 22x2 + 13x + 15. PSHS Main Campus () Theorems on Polynomial Functions July 13, 2012 2/7
3. 3. Factor TheoremFactor TheoremA polynomial function f (x) has a factor (x − a) if and only if f (a) = 0.Examples 1 Conﬁrm that (x − 5) is a factor of x4 − 3x3 + 7x2 − 60x − 125. 2 Show that 2x4 − 11x3 + 14x2 + 9x − 18 is divisible by x2 − 5x + 6. PSHS Main Campus () Theorems on Polynomial Functions July 13, 2012 3/7
4. 4. Rational Zero TheoremRational Root TheoremRZT/RRTIf: 1 f (x) is a polynomial function with integral coeﬃcients, p q, a rational number in simplest terms is a zero of f (x), i.e., 2 p f q = 0,then: 1 p is a factor of the constant term 2 q is a factor of the leading coeﬃcientExampleFind the rational zeros of f (x) = 12x3 − 8x2 − 3x + 2. PSHS Main Campus () Theorems on Polynomial Functions July 13, 2012 4/7
5. 5. Corollary of RZTRZT for an = 1If the leading coeﬃcient of a polynomial function with integral coeﬃcientsis 1, then any rational zeros of f (x) are integers.ExampleFind the rational zeros of f (x) = x4 + 3x3 + 2x2 − 3x − 3. PSHS Main Campus () Theorems on Polynomial Functions July 13, 2012 5/7
6. 6. Fundamental Theorem of AlgebraFundamental Theorem of AlgebraEvery polynomial function with complex coeﬃcients has at least one zeroin the set of complex numbers.Implication of the FTAEvery polynomial function with degree n has exactly n complex zeros.ExampleFind ALL zeros of f (x) = x4 − x3 − x2 − x − 2. PSHS Main Campus () Theorems on Polynomial Functions July 13, 2012 6/7
7. 7. Homework 12 1 Use the remainder theorem to evaluate the functions below: 1 f (−4/5), f (x) = 5x3 − 9x2 + 3x − 11 2 g(1/3), g(x) = 6x3 − 3x2 + 5x − 8 2 Use the factor theorem to determine if the ﬁrst expression is a factor of the second expression. 1 x − 1 ; 2x3 − x2 + 2x − 3 2 4x − 1 ; x3 − 4 x2 + 23 x − 9 2 11 4 3 Find the values of a and b such that x3 − 2ax2 + bx − 3 is divisible by x2 − x − 2. 4 Find a and b such that ax3 − bx2 + 45x + 54 = 0 has 3 as a root and yields a remainder of 12 when divided by x + 1. PSHS Main Campus () Theorems on Polynomial Functions July 13, 2012 7/7