Digging Out and Using Roots to Graph Polynomial Functions Flashcards
What is the degree of that polynominal and why?
3x⁴y⁶ – 2x⁴y – 5xy + 2
The degree is 10. Always take the term which has the highest degree. If the term has two or more variables, add their exponents. Terms are seperated by +- signs.
How many times a quadratic equation can cross the X axis and why?
2 times. If the quadratic equation is a perfect square and you factor it to ()(), you have a maximum of two equations you can solve to find x.
How to proceed if you have this equation?
6x⁴ – 12x³ + 4x²
GCF 2x2(3x2 - 6x + 2)
FOIL (3x**2 - 6x + 2) # this trinominal can´t be
factored, so prime
Please factor!
x² – 2x) + (5x – 10
First step: x(x-2)+5(x-2)
Second step: (x+5)(x-2)
Unfortunalty, you made a little sign mistake during your calculation and are now stuck with this equation. How to proceed?
x(x – 9) + 4(–x + 9)
Just change the middle sign and the signs of the second term.
x(x - 9) - 4(x - 9)
(x-4)(x-9)
Please factor:
1) a³ + b³
2) a³ - b³
3) (a + b)²
4) (a - b)²
5) Solve 25y⁴ – 9
6) Solve 8x³ + 27
1) Sum of cubes (a + b)(a² – ab + b²)
2) Difference of cubes (a – b)(a2 + ab + b2)
3) Sum of squares a² + 2ab + b².
4) Difference of squares (a+b)(a-b)
5)
Rewrite each term as (something)²: (5y²)² – (3)²
Put in formula, so you get: (5y² – 3)(5y² + 3)
6)
Write as sum of cubes: (2x)³ + (3)³
Put in formula: [(2x) + (3)] [(2x)² – (2x)(3) + (3)²].
Simplify: (2x + 3)(4x² – 6x + 9)
Just group, not solve this equation!
4x² – 6x – 6x + 9
Use parentheses and put plus! sign between the terms
4x² – 6x) + (–6x + 9
1) What is the general process to group if you have four or more factors?
2) Show by this example!
x³ + x² – x – 1
1) You group a polynomial into sets with two terms each, and find a GCF in each set.
2)
Break polynomial into sets of two: (x³ + x²) + (–x – 1)
Find GCF of each set and factor out:
for first term: x²(x + 1)
for second term: -1(x + 1)
combine: (x + 1)(x² – 1)
(x² – 1) is a difference of squares and factors again.
You get: (x + 1)(x + 1)(x – 1)
Simplify: (x + 1)²(x – 1).
1) The quadratic equation in standard form is
ax² + bx + c = 0.
- What is the quadratic formula?
- When is it helpful?
2) Solve
x² – 3x + 1 = 0 with it.
1)
- b+-√b^2-4ac) / 2a
- When a quadratic equation just won’t factor anymore.
2)
- First, get equation in right order. Here, it is.
- Then you give the variables their values, from left to right, from a to c. Here: a = 1, b = –3, and c = 1.
- Put it into equation: 3+- SR5
______
2
Solve, and you get the two x intercepts.
Remember! Roots are synonyms for x-intercepts.
1) How many possible roots has
f(x) = 2x⁴ – 9x³ – 21x² + 88x + 48?
2) How many of them are real numbers (positive roots and negative roots)? Which rule to follow?
1) The function can have up to, but no more than 4 roots. You look for the term with the highest degree. This highest degree is the maximum number of possible roots.
2)
Descartes’s rule of signs to find …
… positive real roots: Count how many times the sign changes from term to term. Don´t forget to include first sign!
In this case: 2 times
… negative real roots: change to f(–x) and count again:
f(–x) = 2x⁴ + 9x³ – 21x² – 88x + 48
In this case: 2 times
Thus, there can be at most, but not more than 2 positive real and 2 negative real roots.
What is about roots if a negative number is under the Squareroot?
Then, there is no solution. Thus, the graph won’t cross the x-axis.
How to use the Rational Root Theorem to find all possible rational roots? P.e. 2x+48
- Look at the constant term of the equation, often c.
- Find the factors. For 48, they would be: ±1, ±2, ±3, ±4, ±6, ±8, ±12, ±16, ±24, ±48
- Look at the leading coefficient and its factors. For 2, it would be ±1 and ±2
- Put coefficient factors as denominators, factors of constant term as nominator and list all possibilities: ±1/1, ±2/1, ±3/1, ±4/1, ±6/1, ±8/1, ±12/1, ±16/1, ±24/1, ±48/1, ±1/2, ±2/2, ±3/2, ±4/2, ±6/2, ±8/2, ±12/2, ±16/2, ±24/2, ±48/2
- Simplify: ±1/2, ±1, ±3/2, ±2, ±3, ±4, ±6, ±8, ±12, ±16, ±24, ±48.
Perform the long division of polynominals.
How to write in the end?
Both polynominals should have higher order first. If an exponent is missing, write 0X, 0x2 or whatsoever in there, dor divisor and dividend. During division, when you subtract, you leave these places out, and put them under their right degree. If you have 6x4-30x2+24, you write
6x4 +0x3-30x2+0X+24
What is above strich + reminder
_______
divisor (what is left side from strich)
1) What does the Factor Theorem states?
2 Imagine you have found the three roots 1/2, –3, 4.
1) If x = c is a root, then (x – c) is a factor. Thus, the function (x) which the root was derived from, has a factor.
If x = –1/2 is the root, then (x – (–1/2)) is the factor, better: (x + 1/2)
If x = –3, then (x – (–3)) is a factor, better: (x + 3).
If x = 4, then (x – 4) is a factor with multiplicity two.
