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October 23, 2009 2.4 Real Zeros of Polynomial Functions Divide using long division. 3x3+5x2+8x+7 by 3x+2 October 23, 2009 Find the remainder when f(x)=x3 - 3x + 4 is divided by x + 2. October 23, 2009 Remainder Theorem: If a polynomial f(x) is divided by x-k, then the remainder is r = f(k). Find the remainder when f(x)=3x2+ 7x - 20 is divided by x + 4. October 23, 2009 Q: What does it mean to get a remainder of 0? Factor Theorem: A polynomial function f(x) has a factor of x - k iff f(k)=0. Is x - 3 a factor of x3- x2- x- 15? October 23, 2009 Synthetic Division 3 2 Divide 2x - 3x - 5x - 12 by x - 3 using synthetic division. October 23, 2009 Rational Zeros Theorem: If there are rational zeros in the polynomial: n n-1 f(x) = anx + an-1x + ... + a0 then they will be in the set p q where p = factors of the constant, a0 and q = factors of the leading term coefficient, an October 23, 2009 Find the rational zeros of f(x) = 3x3 + 4x2 - 5x - 2 October 23, 2009 Upper and Lower Bound Tests for Real Zeros Suppose f(x) is divided by x - k using synthetic division. 1. If k≥ 0 and every number in the last line is nonnegative (pos. or zero), then k is an upper bound for the real zeros of f. 2. If k ≤ 0 and the numbers in the last line alternate nonnegative and nonpositive, then k is a lower bound for the real zeros of f. October 23, 2009 Prove that k = 5 is an upper bound for the real zeros of f(x) = 2x3 - 5x2 - 5x - 1 Prove that k = -4 is a lower bound for the real zeros of f(x) = 3x3 - x2 - 5x - 3