Ability to connect experience To with solving quadratic equations from Algebra I to situations analyzing the discriminant will Help students to determine if a quadratic equation has complex roots by reveal the nature of the looking at the graph( the graph does not intersect the x-axis) or the value solutions which would include of the discriminant ( if ) complex solutions N.CN.8 (+) Extend polynomial Knowledge that a negative identities to the complex number can be thought of as numbers. For example, rewrite the square of an imaginary number N.CN.9 (+) Know the Knowledge of the connection The Fundamental Theorem of Algebra states polynomial equation of degree n with complex coefficients has n Fundamental Theorem of between the number of roots “Every roots in the complex numbers. “ Algebra; show that it is true for and the degree of the Factor algebraically, then use the graphing calculator to show the quadratic polynomials. polynomial; considering relationship to the roots of the polynomial. multiple roots, complex roots For example, have students make connections between the zeros of the and distinct real roots polynomials and the behavior of the graph at these values. y = -3x3 + 24x2 45x – y = -3x(x 3)(x 5) the graph of this polynomial – – intersect the x-axis at x=0, x=3 and x=5 y = x4 + 6x3 + 9x2 y = x2(x + 3)2 the of this polynomial will be tangent to the x-axis at x=0 and x = -3 because they are double roots Emphasis should be placed on the ability to solve all types of quadratic polynomials, including those with irrational and complex solutions. Cluster Note: Extend to polynomial and rational expressions.
Ability to connect experience with solving quadratic equations from Algebra I to situations analyzing the discriminant will reveal the nature of the solutions which would include complex solutions To Help students to determine if a quadratic equation has complex roots by looking at the graph( the graph does not intersect the x-axis) or the value of the discriminant ( if ) N.CN.8 (+) Extend polynomial identities to the complex numbers. For example, rewrite Knowledge that a negative number can be thought of as the square of an imaginary number N.CN.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Knowledge of the connection between the number of roots and the degree of the polynomial; considering multiple roots, complex roots and distinct real roots The Fundamental Theorem of Algebra states “Every polynomial equation of degree n with complex coefficients has n roots in the complex numbers. “ Factor algebraically, then use the graphing calculator to show the relationship to the roots of the polynomial. For example, have students make connections between the zeros of the polynomials and the behavior of the graph at these values. y = -3x3 + 24x2 – 45x y = -3x(x – 3)(x – 5) the graph of this polynomial intersect the x-axis at x=0, x=3 and x=5 y = x4 + 6x3 + 9x2 y = x2(x + 3)2 the of this polynomial will be tangent to the x-axis at x=0 and x = -3 because they are double roots Emphasis should be placed on the ability to solve all types of quadratic polynomials, including those with irrational and complex solutions. Cluster Note: Extend to polynomial and rational expressions.
day_1_hs_afternoon_algebra_ii_unit_1_plan.docx