12-02: Working with Polynomials

negative number

The difference between= and an inequality: If we multiply or divide by a(n) ________ in an inequality, we must flip or reverse the inequality.

Process of long division

1. divide 2. multiply 3. subtract 4. bring down

Quotient form

dividend/divisor = quotient + remainder/divisor

Check form

dividend = divisor (quotient) + remainder

Remainder theorem

when polynomial p(x) is divided by (x-b) -> p(b) = R ; when polynomial p(x) is divided by (ax-b) -> P(b/a) = R

(a+b)²

a² + 2ab + b²

(a-b)²

a² - 2ab + b²

a² - b²

(a+b)(a-b)

a³ - b³

(a-b)(a²+ab+b²)

a³ + b³

(a+b)(a² - ab + b²)

Factor theorem

x-b is a factor of a polynomial p(x) if and only if → p(b)=0; ax-b is a factor of a polynomial p(x) if and only if → p(b/a) = 0

Integral zero theorem

If x-b is a factor of polynomial p(x) with leading coefficient 1 and remaining coefficients that are integers then b is a factor of the constant

Rational zero theorem

f polynomial p(x) has integer coefficients and x = b/a is a rational zero of p(x), then b is a factor of the constant, a is a factor of the LC, ax-b is a factor, Possible factors (test values) are any combination of ± b/a

Steps to solving problems where you need to use factors and test values to factor

1. Find factors of a (LC) and b (Constant), both positive and negative (±)

2. Test p(b/a) until we get zero as a result of plugging (b/a) into the p(x) original equation → this is a factor, and we need to change it into an “unsolved” equation (e.g. x=1 would be (x-1) if you unsolved it, and thus your factor)

3. Divide by our factor using long division or synthetic division, you must get a remainder of 0

4. Take the factor found in step (2) and the quotient found in step (3) and smoosh them together

5. Factor fully from both brackets, and solve if needed

Finding the real roots

The real roots of a polynomial equation can be found by setting the equation to zero, y=0
If the polynomial is factorable, the roots can be determined by factoring first fully and then setting each individual factor bracket to zero
1. Factor 2. Solve each root

Families of polynomial functions

A group of functions with common characteristics (e.g. same x intercepts could be a common characteristic) - a value is different for each member of the family

Inequalities

When the = sign is replaced with a

Solving with inequalities

1. Isolate for x 2. Treat the inequality almost like an equal sign. The difference between = and an inequality: If we multiply or divide by a negative number in an inequality, we must flip or reverse the inequality

Interval tables

1. Factor

2. Solve the brackets

3. Draw a number line to identify the intervals

4. Create a chart

• Place intervals in the chart as columns

• Place factors in the chart as rows

5. Place a final function sign row at the bottom

6. Pick a random number in between the intervals (not including the written numbers as those aren’t included, we used circle brackets) and substitute it into the variable in each factor row put in → write the sign of the number that you get

7. Determine the final sign by multiplying all of the signs down to get the overall functions sign

• 2 negatives make a positive, 2 positives make a positive, a mix of negative and positive makes a negative

8. Write your final X E statement based on what your looking for, and remember to put U in between intervals if there is more than one