What is the rate of change of the linear relationship modeled in the table? (4 points) x y −2 5 −1 4 0 3 1 2 a −2 b −1 c negativ
2 answers:
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Problem 33
Use the distributive property on the side containing parentheses of each expression and compare it to the other side.
a) 3(5a + 3) = 15a + 9 not equal to 15a + 6
b) 2(7b - 2) = 14b - 4 not equal to 14b + 4
c) 5(2c + 3) = 10c + 15 not equal to 7c + 8
d) 3(d + 5/3) = 3d + 5 which is equal to 3d + 5
Answer for problem 33: d)
Problem 34
Use a proportion. Let the unknown number of bowls be x. The proportion is made up of two ratios that are set equal to each other. Set each ratio as a ratio of the number of avocados per bowls of guacamole. 3 avocados per 1 bowl (3/1) equals 17 avocados per x bowls (17/x).
Answer: 17/3 full bowls which is the same as 5 2/3 full bowls.
Answer:
Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term") ... so it says "many terms"
Also they can have one or more terms, but not an infinite number of terms.
These are polynomials:
3x
x − 2
−6y2 − ( 79)x
3xyz + 3xy2z − 0.1xz − 200y + 0.5
512v5 + 99w5
5
(Yes, "5" is a polynomial, one term is allowed, and it can be just a constant!)
These are not polynomials
3xy-2 is not, because the exponent is "-2" (exponents can only be 0,1,2,...)
2/(x+2) is not, because dividing by a variable is not allowed
1/x is not either
√x is not, because the exponent is "½" (see fractional exponents)
But these are allowed:
x/2 is allowed, because you can divide by a constant
also 3x/8 for the same reason
√2 is allowed, because it is a constant (= 1.4142...etc)
Monomial, Binomial, Trinomial
There are special names for polynomials with 1, 2 or 3 terms:
monomial, binomial, trinomial
Variables
Polynomials can have no variable at all
Example: 21 is a polynomial. It has just one term, which is a constant.
Or one variable
Example: x4 − 2x2 + x has three terms, but only one variable (x)
Or two or more variables
Example: xy4 − 5x2z has two terms, and three variables (x, y and z)
What is Special About Polynomials?
Because of the strict definition, polynomials are easy to work with.
For example we know that:
If you add polynomials you get a polynomial
If you multiply polynomials you get a polynomial
So you can do lots of additions and multiplications, and still have a polynomial as the result.
Also, polynomials of one variable are easy to graph, as they have smooth and continuous lines.
Example: x4−2x2+x
x^4-2x^2+x
See how nice and
smooth the curve is?
You can also divide polynomials (but the result may not be a polynomial).
Degree
The degree of a polynomial with only one variable is the largest exponent of that variable.
Example:
4x3-x-3 The Degree is 3 (the largest exponent of x)
For more complicated cases, read Degree (of an Expression).
Standard Form
The Standard Form for writing a polynomial is to put the terms with the highest degree first.
Example: Put this in Standard Form: 3x2 − 7 + 4x3 + x6
The highest degree is 6, so that goes first, then 3, 2 and then the constant last:
x6 + 4x3 + 3x2 − 7
You don't have to use Standard Form, but it helps.
