Search the California Content Standards

Cluster:
Visualize relationships between two-dimensional and three-dimensional objects.

Standard:
Verify experimentally that in a triangle, angles opposite longer sides are larger, sides opposite larger angles are longer, and the sum of any two side lengths is greater than the remaining side length; apply these relationships to solve realworld and mathematical problems. CA

Cluster:
Perform arithmetic operations on polynomials. [Beyond quadratic]

Standard:
Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Cluster:
Understand the relationship between zeros and factors of polynomials.

Standard:
Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).

Cluster:
Understand the relationship between zeros and factors of polynomials.

Standard:
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.

Cluster:
Use polynomial identities to solve problems.

Standard:
Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x^2 + y^2)^2= (x^2 – y^2)^2 + (2xy)^2 can be used to generate Pythagorean triples.

Cluster:
Use polynomial identities to solve problems.

Standard:
(+) Know and apply the Binomial Theorem for the expansion of (x + y)^n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.

Footnote:
The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.

Cluster:
Rewrite rational expressions. [Linear and quadratic denominators]

Standard:
Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

Cluster:
Rewrite rational expressions. [Linear and quadratic denominators]

Standard:
(+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

Cluster:
Construct and compare linear, quadratic, and exponential models and solve problems.

Standard:
For exponential models, express as a logarithm the solution to ab^ct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. * [Logarithms as solutions for exponentials]

Cluster:
Construct and compare linear, quadratic, and exponential models and solve problems.

Standard:
Prove simple laws of logarithms. CA *