Factoring Polynomials and Solving Flashcards
When factoring by grouping, what is the first step?
Split the expression into two parts.
Step two: what do the two parenthesis need to do in order to continue factoring?
Match
What are the matching parts considered to be?
The GCF
If the first term is positive what is the GCF?
The GCF is positive.
If the first term is negative what is the GCF?
The GCF is negative.
Step two: if the two parentheses don’t match what two things can you do?
- Look for an error.
2. It cannot factor.
What is used to check the factored expressions?
FOIL
To be sure that you are done factoring, what do you look for?
Difference of perfect squares
When factoring a sum or a difference of cubes, what is the phrase to remember?
Crummy crummy SOPS
Crummy crummy SOPS: crummy crummy =
cube root, cube root
Crummy crummy SOPS: S =
square
Crummy crummy SOPS: OP =
opposite product
Crummy crummy SOPS: second S =
square
What is this formula used for:
a^3 + b^3 = (a+b) (a^2 - ab + b^2)
OR
a^3 - b^3 = (a-b) (a^2 + ab + b^2)
Factoring a sum or difference of cubes
When solving polynomial equations, what tells how many solutions there are?
The biggest exponents.
When factoring in quadratic form, what do you do if the exponents are twice as much (for example ax^4 + bx^2 + c)
Nothing, don’t worry.
How do you check to see if you are done factoring?
FOIL
and make sure to look for a difference of squares.
Check for ____ _____ before circling final answer.
Cube roots
If an equation cannot factor or square root, what are the two other options for solving?
Graph or quadratic formula.
If it doesn’t hit the x-axis it doesn’t have any ____ _________.
real solutions
If there are no real solutions, there are
imaginary solutions
Because you can’t graph points that don’t hit the x-axis, to solve a problem with imaginary solutions use
quadratic formula.
Polynomial vocabulary: one term; a number or a variable.
Monomial
Polynomial vocabulary: the exponent.
Degree of a monomial
Polynomial vocabulary: one or more terms connected by plus or minus.
Polynomial
Polynomial vocabulary: the greatest exponent.
Degree of a polynomial
Polynomial vocabulary: In order with the largest exponent first, then descending.
Standard form of a polynomial
Polynomial vocabulary: used to describe and classify polynomials with two terms.
Binomial
Polynomial vocabulary: used to describe and classify polynomials with three terms.
Trinomial
Polynomial vocabulary: the number before a variable.
Coefficient
Polynomial vocabulary: first coefficient in standard order.
Leading coefficient
Polynomial vocabulary: the term that doesn’t change with the variable; doesn’t have a variable; always comes last in standard order.
Constant
Classifications: degree 0
Constant
Classifications: degree 1
Linear
Classifications: degree 2
Quadratic
Classifications: degree 3
Cubic
Classifications: degree 4
Quartic
Classifications: degree 5
Quintic
Polynomial vocabulary: describes where the graph is going; how the arrows point.
End behavior
Polynomial vocabulary: where the graph changes direction.
Turning points
Hint: the number of turning points is ___ ____ than the degree of the function.
one less
If
Degree: odd
LC: positive
what is the end behavior?
L: falls
R: rises
If
Degree: odd
LC: negative
what is the end behavior?
L: rises
R: falls
If
Degree: even
LC: positive
what is the end behavior?
L: rises
R: rises
If
Degree: even
LC: negative
what is the end behavior?
L: falls
R: falls
Given a #z and a polynomial (P) in standard form, x - z is a factor of the polynomial
If one is true then they all are.
Given a #z and a polynomial (P) in standard form, z is a zero of the polynomial function
If one is true then they all are.
Given a #z and a polynomial (P) in standard form, z is a root/solution of the polynomial equation P(x)=0
If one is true then they all are.
Given a #z and a polynomial (P) in standard form, z is an x-intercept of the graph y=P(x)
If one is true then they all are.
If there is no number before a factor, then
The leading coefficient is zero.
To find the y-intercept from the zeros,
use x=0
If it doesn’t factor,
plug it into the calculator.
What tools on the calculator do you use to find the relative minimum/maximum
CALC
How many relative minimums/maximum can you have?
There is no limit to how many you can have.
To divide two polynomials, use
Polynomial long division
What is the answer to a division problem called?
Quotient
What is the number being divided called?
Dividend
What is the number on the outside of the long division symbol called?
Divisor
What is the number left over after division called?
Remainder
What is another name for synthetic division?
Fake division
Synthetic division steps: 1
Standard form and every lower exponent must be there.
Synthetic division steps: 2
Coefficients go down below the upside-down division symbol.
Synthetic division steps: 3
Add the free zero.
Synthetic division steps: 4
Think: add then multiply
For synthetic long division, if the problem starts with x^4 the answer will start with
x^3
Synthetic long division can only be used for problems with
x + #
and
x - #
Name this process: You list the standard-form coefficients (including zeros) of the polynomial, omitting all variables and exponents. You use “a” for the divisor and add instead of subtract throughout the process.
Synthetic division
What is it if you divide a polynomial P(x) by x-a, then the remainder = P(a)?
Remainder theorem
What can you sometimes use to check a factor and/or help you factor?
Division
From factoring by grouping, if the two terms in parentheses don’t match, what can you do?
Use synthetic division (sneaky factoring).
After using “sneaky factoring” (synthetic division), what do you do if the problem isn’t factored all the way?
Use the box method to factor further.
How do you find the y-intercept from the x’s?
Plug in zero for x and multiply what is left together.
When figuring out which regression to use (linear, quadratic, cubic, quartic, etc.), what do you look for?
The R^2 value to be closest to 1.
What numbers will always have it’s opposite as a factor?
Radical and imaginary roots.
