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Algebra II: Polynomials and Rationals

5 dot points across 5 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How do complex numbers arise from quadratics with negative discriminants, and how do you operate with them?

Define the imaginary unit i and operate with complex numbers (add, subtract, multiply); use the discriminant to determine the nature of a quadratic's roots; and solve quadratics with complex roots using the quadratic formula or completing the square.
A NY Regents Algebra II answer on complex numbers and quadratics: the imaginary unit i, adding/subtracting/multiplying complex numbers, the discriminant and the nature of roots, and solving quadratics with complex solutions.
9 min answer →

How do you divide polynomials, and what does the Remainder Theorem tell you about factors and values?

Divide polynomials using long division and synthetic division; apply the Remainder Theorem (the remainder when dividing by x minus a equals the value of the polynomial at a) and the Factor Theorem to test for factors and find zeros.
A NY Regents Algebra II answer on polynomial division and the Remainder Theorem: long and synthetic division, why the remainder equals the polynomial value, and using the Factor Theorem to confirm factors and zeros.
9 min answer →

How do the zeros, their multiplicities, and the leading term shape the graph of a polynomial?

Find the zeros of a polynomial from its factored form; use multiplicity to decide whether the graph crosses or touches the x-axis; and use the degree and leading coefficient to determine end behavior, then sketch the graph.
A NY Regents Algebra II answer on polynomial graphs: finding zeros from factored form, how multiplicity makes a graph cross or touch the x-axis, and how degree and leading coefficient set the end behavior.
9 min answer →

How do you rewrite radicals as rational exponents, simplify them, and solve radical equations?

Convert between radical and rational-exponent form; simplify radical and rational-exponent expressions using the exponent laws; and solve radical equations, checking for extraneous solutions introduced by squaring.
A NY Regents Algebra II answer on radicals and rational exponents: converting between forms, simplifying with the exponent laws, and solving radical equations while rejecting extraneous solutions from squaring.
9 min answer →

How do you simplify and combine rational expressions, and how do you solve a rational equation while avoiding extraneous solutions?

Simplify rational expressions by factoring and cancelling (noting domain restrictions); add, subtract, multiply, and divide them; and solve rational equations, checking for extraneous solutions introduced by the denominators.
A NY Regents Algebra II answer on rational expressions: simplifying by factoring with domain restrictions, the four operations on rational expressions, solving rational equations, and rejecting extraneous solutions.
9 min answer →