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A2 Concepts, topics, & problems Chapter 6
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Use this to study for your final.
Terms in this set (101)
Monomial vs Polynomial
A monomial is an expression that is a real number, a variable, or a product of real numbers and variables with whole number exponents. A polynomial is a monomial or a sum of them.
A polynomial in standard form cannot have what?
Two terms with the same degree
Write 2m^2 - 3 + 7m in standard form.
2m^2 + 7m - 3
In what two ways can you classify a polynomial?
By degree and number of terms
Names for degrees- 0, 1, 2, 3, 4, 5
Constant, Linear, Quadratic, Cubic, Quartic, Quintic
Names for number of terms- 1, 2, 3, 4, 5
Monomial, Binomial, Trinomial, Polynomial of 4 terms, Polynomial of 5 terms
Classify 5x^2 + x
Quadratic binomial
Classify 6x^4 + 3x^3 + 2x^2 + 5
Quartic polynomial of 4 terms
What fits data more closely than linear and quadratic models?
Polynomial models of degree three or greater
Which option on a graphing calculator do you use to create a cubic model?
CubicReg
How do you estimate a value for y using a graphing calculator?
Use the Table option of a graphing calculator.
What does end behavior describe?
The far left and far right portions of the graph and the directions
How can you determine the end behavior of the graph of a polynomial function on the right side?
If the leading coefficient is positive, the graph rises to the right and if negative, it falls.
How can you determine the end behavior of the graph of a polynomial function on the left side?
If the degree is even. the left is the same as the right, and if the degree is odd, the left is the opposite of the right.
Determine the end behavior of the function -2x^4 + 8x^3 - 8x^2.
On the right side, it falls, and on the left side it also falls.
You can write a polynomial as a product of what?
Linear factors
How is a linear factor similar to a prime number?
A linear factor cannot be factored any further.
Write (x + 1)(x + 2)(x + 3) as a polynomial in standard form.
x^3 + 6x^2 + 11x + 6
Write x(x - 1)(x + 1) as a polynomial in standard form.
...
What can you use to factor a polynomial?
The GCF of the terms
Factor 6x^3 - 15x^2 - 36x.
look at back of textbook
Find the relative minimum, maximum, and zeros of the function f(x) = x^3 + 4x^2 - 5x.
See textbook?
How do you use the Zero Product Property to find the zeros of a polynomial function?
Find a zero for each linear factor, set each linear factor to equal 0 and solve.
What do you use to write the linear factors knowing the zeros?
Factor Theorem
What are the four key facts about a polynomial that the Factor Theorem helps relate?
See page 316 of textbook.
When is a polynomial function in factored form?
Once it has been factored to its linear factors
What can you use to find all the zeros of a polynomial function?
Polynomial Division
What is polynomial division and how do you do it?
Polynomial division is similar to numerical division in that you can test what the factors are. You follow the same procedure as with whole number division.
Divide x^2 + 3x - 12 by x - 3.
x + 6, R 6
Say what each term in division is called in order: 4 / 2 = 2
Dividend, divisor, quotient
What else can you use polynomial long division for?
Find the factors of a polynomial
Determine whether or not x + 4 is a factor of x^2 + 6x + 8.
Yes, it is a factor.
What other simpler way can you divide by a linear factor?
Synthetic Division
What is synthetic division?
A simplified process to divide a polynomial by a linear factor by omiting variables and exponents, reversing the sign in the divisor, and adding instead of subtracting.
Use synthetic division to divide 3x^3 - 4x^2 + 2x - 1 by x + 1.
3x^2 - 7x + 9, R - 10
Use synthetic division to divide x^3 + 4x^2 + x - 6 by x + 1.
Look at back of textbook.
A polynomial divided by x-3 results in another factor with a remainder of 5. What will the remainder be?
P(3), so 5
Find P(-4) for P(x) = x^4 - 5x^2 + 4x + 12 using the remainder theorem and synthetic division.
...
What can you use the Remainder Theorem for?
To find values of P(x)
Use synthetic division to find P(-1) for P(x) = 2x^4 + 6x^3 - 5x^2 - 60.
See back of textbook.
Divide using long division. (2x^3 - 3x^2 - 18x - 8) / (x - 4)
Check online answers.
Determine whether x + 3 is a factor of x^3 + 4x^2 + x - 6.
Check online answers.
Divide (x^3 + 27) / (x + 3) using synthetic division.
Check online answers.
Use synthetic division to factor y = x^3 - 4x^2 - 9x + 36; (x + 3)
Check online answers.
width = x - 2
The volume in cubic inches of a decorative box can be expressed as the product of the lengths of its sides as V(x) = x^3 + x^2 - 6x.
Check online answers.
A polynomial P(x) is divided by x-a. The remainder is zero. What conclusion can you draw?
Check online answers.
A student represented the product of 3 linear factors as x^3 - x^2 - 2x. She used x-1 as one of the factors. Use division to prove that the student made an error.
Check online answers.
Write a polynomial division that has a quotient of x + 3 and a remainder of 2.
I think it is (x + 3)(x - 5) + 2 but check online answers.
What is the remainder when x^2 - 5x + 7 is divided by x + 1?
Check online answers.
Write the polynomials in standard form with the given zeros.
3, -5
0, 1, 8
-1, 2, 5
1, multiplicity 4
Check online answers.
Write a polynomial function with at least three zeros that are negative, one of which has multiplicity 2.
Check online answers.
How can you use graphing to solve a polynomial equation?
Graphing each side of the equation separately and finding the x values at the point (s) of intersection.
Solve x^3 + 3x^2 = x + 3 by graphing.
-3, -1, 1
Solve x^3 - 2x^2 = -3 by graphing. Don't forget to check your answers.
See back of textbook.
Find the dimensions of a carrier with volume 7 ft^3, width x inches, length (x+3) inches, and height (x-2) inches.
Check back of textbook.
What is the factoring pattern for a^3 + b^3 and a^3 - b ^3?
(a + b)(a^2 - ab + b^2) and (a - b)(a^2 + ab + b^2)
Factor 8x^3 - 1.
Check back of textbook.
Solve x^3 + 8 = 0 using the sum and difference of cubes properties.
Check back of textbook.
Solve 27x^3 - 1 = 0 using the sum and difference of cubes properties.
Check back of textbook.
How can you factor a polynomial of higher degree in a like technique to how polynomials of lower degree are solved?
By using a quadratic pattern
How do you use a quadratic pattern to factor a polynomial?
Rewrite the polynomial in the pattern of a quadratic expression, substitute a for x^2. Factor in that form and then substitute x^2 for a. Factor completely.
Factor x^4 + 7x^2 + 6 using a quadratic pattern.
Check back of book.
Factor x^4 - 3x^2 - 10 using a quadratic pattern.
Check back of book.
Solve x^4 + 11x^2 + 18 = 0.
Check back of book.
Solve 2x^4 - 5x^3 - 3x^2 = 0 by graphing. Don't forget to check answers.
Check back of book.
Factor x^3 - 1000.
Check back of book.
Solve 8x^3 - 1 = 0.
Check back of book.
Factor x^4 + 8x^2 - 20.
Check back of book.
Solve x^3 + 3x^2 - 4x - 12 = 0.
Check back of book.
Solve 81x^3 - 192 = 0.
Check back of book.
With rational roots, the numerators of the roots are factors of what?
The constant term
With rational roots, the denominators of the roots are factors of what?
The leading coefficicent
How can you use the Rational root theorem?
To find any rational roots of a polynomial equation with integer coefficients and to find all of the roots of a polynomial equation
Find the rational roots of x^3 - 4x^2 - 2x + 8 = 0.
Check
Find the roots of x^3 - 2x^2 - 5x + 10 = 0.
check
Find the roots of 3x^3 + x^2 - x + 1 = 0.
...
What can you use conjugates for?
To find the irrational roots of a polynomial equation
A polynomial equation with rational coefficients has the roots 2 - √-7 and √5. Find two additional roots.
Check.
One of the roots of a polynomial equation is 4 - √2. Can you be certain that 4 + √2 is also a root of the equation? Explain.
Check.
What can you use complex conjugates for?
To find an equation's imaginary roots
If a polynomial equation with real coefficients has 3i and -2 + i among its roots, then what two other roots must it have? Then describe the degree of the equation.
Check bob.
What can help write a polynomial equation?
The Irrational Root Theorem and the Imaginary Root Theorem
Find a third-degree polynomial equation with rational coefficients that has roots -1 and 2 - i.
Check bob
Find a fourth- degree polynomial equation with rational coefficients that has roots i and 2i.
Check bob
List all possible rational roots of 2x^3 - 9x^2 - 11x + 8 = 0 and then find any actual rational roots.
check
Find the roots of 4x^3 + 16x^2 - 22x - 10 = 0.
check
Find two additional roots for the polynomial equation with given roots √5 and -√13
check
Find two additional roots for the polynomial equation with given roots 2 and -4i
check
#32-47 pg 339
check
How many roots does x^3 have? How about linear factors?
3, 3
Find the number of complex roots, the possible real roots, and the possible rational roots, using the theorems, of x^4 - 3x^3 + x^2 - x + 3 = 0.
back of book
How can you find all the zeros of a polynomial functions?
Using a combination of graphing, the factor theorem, polynomial division, the remainder theorem, and the quadratic formula
Find all zeros of y = x^3 - 2x^2 + 4x - 8. Which zeros can you verify on a graphing calculator?
check back of book
Find the number of complex roots, the possible real roots, and the possible rational roots of 4x +8 = 0 and -2x^6 - x^2 + x - 7 = 0.
online answers?
Find all zeros of f(x) = x^3 - 3x^2 + x - 3 and g(x) = x^3 - 1/2x^2 + 20x - 10.
online answers?
Write a polynomial function that has four possible rational zeros but no actual rational zeros.
online answers?
Using the Rational Root Theorem, explain why the following statement is false: if a polynomial has no constant term, then the corresponding polynomial equation has only the number 0 as a possible rational root.
online answers?
Consider a polynomial with integer coefficients in which the leading coefficient is not equal to 1. Can the related polynomial equation have a rational root that is an integer?
online answers?
#32- 39 on pg 344
check
Do chapter 6 review and test
check
Solve x^4 - 14x^2 + 49 = 0 by factoring.
check
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