So 5 is another zero. Use the Quadratic Formula on the depressed polynomial 2x + 4 to find the 1 zeros, – – 1 ± . i The function has two real zeros at 3 and and two imaginary zeros at 1 ± . i –5 Exercises Find all zeros of each function. 1. = + + 9x + 9 2. = 3 + 4x 12 f(x) f(x) – – 3. = 10 + 34a 40 4. = 5 + 11x 15 p(a) p(x) – – – – 5-8 Study Guide - Rational Zero Theorem Identify Rational Zeros Let f (x) = + + + + x + represent a polynomial function with integral coefficients. If is … Rational Zero Theorem a rational number in simplest form and is a zero of y = f(x), then p is a factor of and q is a factor of . Corollary (Integral Zero If the coefficients of a polynomial are integers such that = 1 and 0, any rational zeros of ≠ the function must be factors of . Theorem Example : List all of the possible rational zeros of each function. a. = 3 2 + 6x 10 f(x) – – If is a rational root, then is a factor of and is a factor of 3. The possible values for are ±1, ±2, ±5, and ±10. p q p –10 The possible values for are 61 and 63. So all of the possible rational zeros are = ±1, ±2, ±5, ±10, ± , ± , ± , and ± q . b. = 10 + 14x 36 q(x) – – Since the coefficient of is 1, the possible rational zeros must be the factors of the constant term So the possible x3 –36. ±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18, ±36. rational zeros are and Exercises List all of the possible rational zeros of each function. 1. = + 3 + 8 2. = 7 + 3 + 20 f(x) x g(x) x – – – 3. = 7 4 + 49 4. = 2 5 + 8 + 3x 5 h(x) – – x – p(x) – –
So – 5 is another zero. Use the Quadratic Formula on the depressed polynomial – 2x + 4 to find the 1 zeros, 1 ± i . The function has two real zeros at 3 and –5 and two imaginary zeros at 1 ± i . Exercises Find all zeros of each function. 1. f(x) = + + 9x + 9 2. f(x) = – 3 + 4x – 12 3. p(a) = – 10 + 34a – 40 4. p(x) = – 5 + 11x – 15 5-8 Study Guide - Rational Zero Theorem Identify Rational Zeros Rational Zero Theorem Let f (x) = + + … + + x + represent a polynomial function with integral coefficients. If is a rational number in simplest form and is a zero of y = f(x), then p is a factor of and q is a factor of . Corollary (Integral Zero Theorem If the coefficients of a polynomial are integers such that = 1 and ≠ 0, any rational zeros of the function must be factors of . Example : List all of the possible rational zeros of each function. a. f(x) = 3 – 2 + 6x – 10 If is a rational root, then p is a factor of –10 and q is a factor of 3. The possible values for p are ±1, ±2, ±5, and ±10. The possible values for q are 61 and 63. So all of the possible rational zeros are = ±1, ±2, ±5, ±10, ± , ± , ± , and ± . b. q(x) = – 10 + 14x – 36 Since the coefficient of x3 is 1, the possible rational zeros must be the factors of the constant term –36. So the possible rational zeros are ±1, ±2, ±3, ±4, ±6, ±9, ±12, ±18, and ±36. Exercises List all of the possible rational zeros of each function. 1. f(x) = + 3 – x + 8 2. g(x) = – 7 + 3 + x – 20 3. h(x) = – 7 – 4 + x – 49 4. p(x) = 2 – 5 + 8 + 3x – 5
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