Secondary Math 3

2.3 Real and Non Real Roots of a PolynomialPolynomial IdentitiesSecondary Math 3WARM UPGiven the polynomial What is the degree?What is the end behavior?What are the zeros?Sketch a graph.

Complex ZerosThe Fundamental Theorem of Algebra states that a polynomial of nth degree will have n complex zeros.

Complex zeros (a + bi) can be either real numbers or non real (imaginary) numbers.

When roots of a polynomial include non real numbers (the graph will not cross the x-axis) it will always include a non real number and its conjugate (a + bi and a bi).

Thus, non real roots come in pairs.Three polynomials of degree 4 are graphed below. Describe the roots.r(x) has four real rootsq(x) has two real roots and two non real rootsp(x) has four non real roots

Three polynomials of degree 3 are graphed below. Describe the roots.r(x) one real root and two non real rootsq(x) has three real rootsp(x) has three real roots

QuestionsA polynomial has a degree of 8. Which of the following could be the number of real roots? 2, 3, 4, 7, or 8

A polynomial with real coefficients has a degree of 5. Which of the following could be the number of complex non-real roots? 2, 3, 4, 6, or 7

Assume that the degree of a polynomial is odd. The number of real roots would be even or odd?

Polynomial IdentitiesPerfect Square TrinomialDifference of Squares

Polynomial IdentitiesCubic Polynomials

Polynomial IdentitiesSum of CubesDifference of Cubes

Polynomial IdentitiesTrinomial Leading Coefficient of 1Sum of Squares

Proving Polynomial IdentitiesTo prove an identity you simplify or change one side to get the other side.

Quadratic Formula ProofSolve by completing the square.

Irreducible or Prime PolynomialA polynomial with integer coefficients that can not be factored into polynomials of lower degree, also with integer coefficients.Examples Multiply using Polynomial Identities

Examples Multiply using Polynomial Identities

Examples Factor expressions using the Polynomial Identities.

Examples Factor expressions using the Polynomial Identities.