Precalculus (Unit 4)

When finding possible real zeros…

Imaginary roots come in pairs! Don’t forget to list the answer in decrements of 2! (6, 4, 2, or 0)

Write out f(-x) for sign changes!

When factoring completely…

Always check a value! If you used integer coefficients you likely won’t need one but if you have any fractions (Like in an irrational root) be sure to simply take the original a value, divide it by the denominators of the fractions you already removed, and then write it! Or just find out what the current a value is and add the number that gives the correct value ( 3(x+3)(3x+2) when I need 9!)

Synthetic Division tips

• Don’t haphazardly flip the signs when adding the product to the next coefficient

• If there is a “gap” (x^4+x²+x) the missing coefficient will just be zero!

theory of coefficients

• leading coefficient must be one

• + - + - + - + -…for each step

• sum, sum of pairs, sum of triples, sum of quadruples, etc. but always end in product of zeros

• patterns: quartics will have 6 pairs, 4 triples

All factoring strategies

• Synthetic Division

• P/Q (only if decartes says there are zeros)

• Grouping

• Splitting middle term (leading co. * constant, what 2 numbers multiply to it and add to OG middle term)

• Basic Method (split leading co. into two factors) (2x+ )(2x+ ) (be careful, if this doesn’t work use quad as always)

• If you have a x^4+x²+#, use basic

When is theory of coefficients invoked?

when you need to make a standard form equation from a set of zeros

What is a common mistake with theory of coefficients?

Make sure the signs are correct! Always double check the component before you write it down!

How does the leading exponent relate to the limits?

Even = both will be the same

Odd = both will be opposites

For test day: How do you write p/q?

List all the possible zeros! Don’t write +-(fraction)

sum and difference of cubes

(a+b)(a²-ab+b²)

(a-b)(a²+ab+b²)

where 27x³ = a³x³ (27 is not a, but a³)

Emergency factoring strategy? (like with big numbers)

Brute force with quad formula (do all steps by hand, multiplying and all)

More mistakes (with writing functions from graphs and finding y-ints)(and factoring)

• Don’t forget the exponent of the factor when finding the y-int!

• Double check for arithmetic errors

• Don’t forget to completely factor squares and all! (x²-4) becomes (x+2)(x-2)

Remainder Theorem (could save time)

the remainder of division is the output of that number

-2 has a remainder of 6, so f(-2) = 6

therefore, you could go about certain problems by just plugging in -2 and seeing if it gets you zero

What to include in inequality answers

• List of zeros

• Completely factored equation

• number line

• range of solutions in interval notation

When doing theory of coefficients…

just write out all steps! It doesn’t take that long!!!

ESPECIALLY on sum of pairs! Be careful and don’t accidentally multiply anything by itself!

if you need to make integer coefficients…

multiply by the largest denominator! (1/36 vs ¼, choose 36)

Test Corrections

• Remember to get the concrete number of roots from the leading degree and combine this knowledge with the number of given roots

  • For example, if you have 5 possible roots but given 4 non-reals, then the fifth one must be real as non-reals come in pairs

• Concepts: If # is a zero of a function, then plugging that number into the function will get you zero