7-5 Roots and Zeros 7-6 Rational Zero Theorem

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7-5 Roots and Zeros 7-6 Rational Zero Theorem Skills: Determine the number and type of roots for a polynomial equation. Find the zeros of a polynomial function. Identify the possible rational zeros of a polynomial function. Find all rational zeros of a polynomial function. Glencoe – Algebra 2 Chapter 7: Polynomial Functions

Fundamental Theorem of Algebra Corollary Fundamental Theorem of Algebra If f(x) is a polynomial of degree n where n>0, then the equation f(x)=0 has exactly n solutions, including doubles/triples. If f(x) is a polynomial of degree n where n>0, then the equation f(x)=0 has at least one complex solution. Behavior Near Zeros Complex Conjugates Theorem When a factor x – k is raised to an odd power (single/triple zero), the graph crosses the x-axis at x = k. When a factor x – k is raised to an even power (double/quadruple zero), the graph touches (is tangent to) the x-axis at x = k. If a + bi is a zero, then so is its conjugate a – bi. Irrational Conjugates Theorem If is a zero, then so is its conjugate . Descartes’ Rule of Signs Positive Real Zeros: # of sign changes in f(x) or less by an even number Negative Real Zeros: # of sign changes in f(-x) or less by an even number Glencoe – Algebra 2 Chapter 7: Polynomial Functions

Rational Zero Theorem (Rational Roots Theorem) If the polynomial has integer coefficients, then all of the rational roots will be of the form: Finding Rational Roots (Zeros) Use the rational zero theorem to find all possible roots. Test the possible roots by plugging them into the equation until you get one root. If the answer you get is zero, then the number is a root. Use synthetic division to break the polynomial down into factors. Solve using the factor theorem to find the roots. If you’re having trouble factoring, test other possible roots in the polynomial you’re trying to factor again and use synthetic division. Glencoe – Algebra 2 Chapter 7: Polynomial Functions

Steps for Finding All Zeros Use Descartes’ Rule of Signs to make a zeros table. Use the Rational Zeros Theorem to find all possible rational zeros. Use synthetic division to test the zeros and break down the polynomial. If you can, factor. Use the quadratic formula to find irrational/imaginary zeros. Remember irrational/imaginary zeros come in pairs (conjugates). Glencoe – Algebra 2 Chapter 7: Polynomial Functions

Chapter 7: Polynomial Functions Example 1 Find all zeros of 1 2 3 1 2 Positive Negative Imaginary Total 3 2 5 1 4 Glencoe – Algebra 2 Chapter 7: Polynomial Functions

Chapter 7: Polynomial Functions Example 2 Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the following zeros: . Glencoe – Algebra 2 Chapter 7: Polynomial Functions

Example 3 Find the rational zeros of Glencoe – Algebra 2 Chapter 7: Polynomial Functions

Chapter 7: Polynomial Functions Example 4 Find the real zeros of Glencoe – Algebra 2 Chapter 7: Polynomial Functions