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7.6 Rational Zero Theorem. Objectives: Identify the possible rational zeros of a polynomial function. Find all the rational zeros of a polynomial function. Rational Zero Theorem and Corollary.
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7.6 Rational Zero Theorem Objectives: • Identify the possible rational zeros of a polynomial function. • Find all the rational zeros of a polynomial function.
Rational Zero Theorem and Corollary • Let f(x) = a0xn + a1xn-1+…+an-1x + an represent a polynomial function with integral coefficients. If p/q is a rational number in simplest form and is a zero of y = f(x), then p is a factor of an and q is a factor of a0. • Example: • Given 4x3 + 3x2 – 7x + 18. If 9/2 is a zero of f(x), then 9 is a factor of 18 and 2 is a factor of 4. • Corollary (Integral Zero Theorem) • If the coefficients of a polynomial function are integers such that a0 = 1 and an≠ 0, any rational zeros of the function must be factors of an.
In terms you can understand! • Given f(x) = x4 + 6x3 + 7x2 – 6x – 8. Find the possible rational roots. • The only rational roots are factors of 8 (an) divided by factors of 1(a0). • an = -8. so the factors are ±1, ±2, ±4, ±8 • a0 = 1, so the factors are ± 1. • So if f(x) HAS rational roots, the only ones can be ±1, ±2, ±4, ± 8
Another problem • Find the possible rational roots of • f(x) = 6x4 + 2x3 – x2 + 7x – 4 • an = -4 ,so factors are ±1, ±2, ±4 • a0 = 6, so factors are ±1, ±2, ±3, ±6 • So all possible rational zeros are the factors of an: ±1, ±2, ±4, and p/q: ±1/2, ± 1/3, ± 1/6, ± 2/3, ± 4/3
Find zeros • Given f(x) = 2x4 – 5x3 – 4x2 + 10x – 3, find all the zeros. • Using Descarteś Rule of signs: • Positive real roots, 3 or 1. • f(-x) = 2x4 + 5x3 – 4x2 – 10x – 3 • Negative real roots, 1. • a0 = 2, factors: 1, 2 an = -3, factors;1,3 • So my possible rational roots are • ±1, ±3, ±1/2, ±3/2
continued f(x) = 2x4 – 5x3 – 4x2 + 10x – 3 we know to test ±1, ±3, ±1/2, ±3/2 using synthetic division 2x³-3x²-7x+3 Start over
continued Solve remaining polynomial 2x²-6x+2=0 Roots are: 1, -3/2,
Find all zeros. • h(x) = 10x3 – 17x2 – 7x + 2 • D ROS gives us 2 or 0 positive real roots and 1 negative real root. • Our possible roots are • ±1, ±2, ±1/2, ± 1/5, ± 1/10, ±2/5 • We either have 2 positive and 1 negative real roots or 2 imaginary and 1 negative real. • Find out what they are!
