Preliminary information Algebra 2

What do successive differences help you find?

The degree of a polynomial sequence.

What do you do if differences are not constant?

Keep taking differences until they become constant.

For a quadratic sequence, what is the second difference?

2a.

For An^k, how do you find A from the kth difference d?

A = d/k!.

What numbers do you look for when factoring x^2+bx+c?

Two numbers that multiply to c and add to b.

What is the polynomial division relationship?

Dividend = divisor × quotient + remainder.

What is (a+b)^2?

a^2+2ab+b^2.

What is (a-b)^2?

a^2-2ab+b^2.

What is a^2-b^2?

(a-b)(a+b).

What is a^3+b^3?

(a+b)(a^2-ab+b^2).

What is a^3-b^3?

(a-b)(a^2+ab+b^2).

What is (a+b)^3?

a^3+3a^2b+3ab^2+b^3.

What is (a-b)^3?

a^3-3a^2b+3ab^2-b^3.

What shortcut can multiply numbers equally spaced around a middle number?

Middle^2-distance^2.

What identity does that shortcut use?

(x+a)(x-a)=x^2-a^2.

What is a Mersenne number?

A number of the form 2^p-1.

What must p be for 2^p-1 to possibly be prime?

p must be prime.

Does prime p guarantee 2^p-1 is prime?

No.

What equation defines a Pythagorean triple?

a^2+b^2=c^2.

How do you generate a Pythagorean triple?

a=x^2-y^2, b=2xy, c=x^2+y^2.

What does Pascal's Triangle provide?

Binomial coefficients.

What does the Binomial Theorem expand?

Powers of a binomial.

How do exponents change in a binomial expansion?

First decreases; second increases.

What do the exponents in each term add to?

n, the original exponent.

What is C(n,k)?

n!/[(n-k)!k!].

What is a permutation?

An arrangement where order matters.

What is a combination?

A selection where order does not matter.

What is the permutation formula?

P(n,r)=n!/(n-r)!.

What is the combination formula?

C(n,r)=n!/[r!(n-r)!].

What question distinguishes permutations from combinations?

Does order matter?

What is multiplicity?

How many times a zero occurs as a factor.

What happens at an odd-multiplicity zero?

The graph crosses the x-axis.

What happens at an even-multiplicity zero?

The graph touches and turns around.

How do you find the degree from factored form?

Add the exponents of the factors.

What is the maximum number of distinct real zeros for degree n?

n.

How many complex roots does degree n have?

n, counting multiplicity.

What is the maximum number of relative extrema for degree n?

n-1.

What does the Remainder Theorem say?

Dividing f(x) by x-a gives remainder f(a).

What do you substitute for x when dividing by x-a?

a.

What do you substitute for x when dividing by x+a?

-a.

How do you check whether x-a is a factor?

Check whether f(a)=0.

What do you substitute when dividing by ax+b?

-b/a.

Why is the Remainder Theorem useful?

It can replace long division with substitution.

What is the first step when completing the square?

Move the constant to the other side.

How do you find the number needed to complete the square?

Divide the x coefficient by 2 and square it.

What form do you create when completing the square?

A perfect-square trinomial.

What is the quadratic formula?

x=[-b±sqrt(b^2-4ac)]/(2a).

When can you use the quadratic formula?

To solve ax^2+bx+c=0.

What restriction applies to a rational-expression denominator?

It cannot equal 0.