If a polynomial q(x) is divided by x – 4, the quotient is 2 𝑥 2 +3𝑥+11+ 58 𝑥−4 . Which of the following must be true? q(-4) = 58 q(4) = 58 q(-4) does not exist q(4) does not exist Problem of the Day
Problem of the Day… Take 2! Given that p(-2) = 0, p(-5) = 0, and p(4) = 0, which expression could be p(x)? x 3 +3 x 2 −18x−40 x 3 −3 x 2 +18x−40 x 3 −3 x 2 −18x+40 x 3 −3 x 2 −18x−40 Problem of the Day… Take 2!
Section 5-7 Roots and Zeros
Then Now Objectives You used complex numbers to describe solutions of quadratic equations. Determine the number and type of roots for a polynomial equation. Find the zeros of a polynomial function.
Common Core State Standards Content Standards N.CN.9 – Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. A.APR.3 – Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. Mathematical Practices 6) Attend to precision. Common Core State Standards
Zeros, Factors, Roots, and Intercepts
Find all of the zeros of: 𝑓 𝑥 = 𝑥 3 +2 𝑥 2 +9𝑥+18 Example 3
Find all of the zeros of: 𝑔 𝑥 = 𝑥 4 +50 𝑥 2 +49 Example 3
Find all of the zeros of: ℎ 𝑥 = 𝑥 4 −3 𝑥 3 − 𝑥 2 −27𝑥−90 Example 3
Find all of the zeros of: 𝑓 𝑥 = 𝑥 4 −3 𝑥 3 −9 𝑥 2 +77𝑥+150 Example 3
Complex Conjugates Theorem
Write a polynomial function of least degree with integral coefficients, the zeros of which include -5, 4i. Example 4
Write a polynomial function of least degree with integral coefficients, the zeros of which include 6, -2i. Example 4
Write a polynomial function of least degree with integral coefficients, the zeros of which include -1, 4, 3i. Example 4
p.363 #11, 15, 35, 38, 41, 42, 50 – 52, 61 Homework
Which of the following functions has only real zeros? a) b) c) d) Problem of the Day