Answer:
[tex]f(x)=x^3-2 x^2+9 x-18[/tex]
Step-by-step explanation:
The roots of the polynomial are [tex]2,3i,-3i[/tex].
This implies that [tex]x-2,x-3i,x+3i[/tex] are factors of the given polynomial.
The polynomial will have equation;
[tex]f(x)=(x-2)(x-3i)(x+3i)[/tex]
We expand using difference of two squares on the complex conjugates to get;
[tex]f(x)=(x-2)(x^2-(3i)^2)[/tex]
[tex]\Rightarrow f(x)=(x-2)(x^2-(-3)^2(i)^2)[/tex].
[tex]\Rightarrow f(x)=(x-2)(x^2-9(i)^2)[/tex].
Recall that;
[tex]\boxed{i^2=-1}[/tex]
[tex]\Rightarrow f(x)=(x-2)(x^2+9)[/tex].
Expand using the distributive property to get;
[tex]\Rightarrow f(x)=x^3+9x-2x^2-18[/tex].
We rewrite in standard form to obtain;
[tex]f(x)=x^3-2x^2+9 x-18[/tex]
angle ABD and angle DBC are supplementary. Find the value of x.
A. 6
B. 8
C. 4
D. 10
Read the following statement: If the sum of two angles is 90°, then the angles are complementary. The hypothesis of the statement is:
there are two angles.
the sum of two angles is 90°.
the angles are complementary.
Angles are complementary if their sum is 90°.
The hypothesis in the given mathematical conditional statement 'If the sum of two angles is 90°, then the angles are complementary.' is 'the sum of two angles is 90°'.
Explanation:In a conditional statement in mathematics, the 'if' part of the statement is called the hypothesis and the 'then' part is termed the conclusion. Given the statement 'If the sum of two angles is 90°, then the angles are complementary.', the hypothesis of this statement is 'the sum of two angles is 90°'.
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In a conditional statement, the hypothesis is the condition that needs to be met. In this case, the hypothesis of the statement 'If the sum of two angles is 90°, then the angles are complementary,' is 'the sum of two angles is 90°.'
Explanation:In the context of the given conditional statement, 'If the sum of two angles is 90°, then the angles are complementary,' the hypothesis refers to the clause immediately after 'if.' This indicates the condition that needs to be fulfilled for the conclusion to be considered valid. Therefore, the hypothesis for this statement is 'the sum of two angles is 90°'.
After the 'if,' the hypothesis is given, and after the 'then,' you find the conclusion. The conclusion in this case is 'the angles are complementary.'
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Four times the sum of a number and 15 is at least 120. Find all possible solutions for x
12.5% of what is 130?
WILL GIVE BRAINLIST!!!!!!!!!!!!!! A group of students were surveyed to find out if they like working as a camp counselor and/or as a lifeguard during summer break. The results of the survey are shown below:
32 students like working as a camp counselor
8 students like working as a camp counselor but do not like working as a lifeguard
29 students like working as a lifeguard
8 students do not like working as a lifeguard or a camp counselor
Four students created the tables below to represent the data. CC represents camp counselor and LG represents lifeguard.
Which student's table is correct?
Answer:
I think Table C - Corralle
Answer:
c
Step-by-step explanation:
If a train travels one mile (5,280 feet) while climbing a hill at an angle of five degrees, approximately how many vertical feet has the train climbed?
to calculate the vertical height multiply the hypotenuse ( 5280) by the sin of the angle (5)
5280 x sin(5) = 460.1823
round off to 460 feet
Find the domain of the following piecewise function
How to factor 2p^4+9p^3-18p^2
A lawn mower manufacturer incurs a total of $34,816 in overhead costs and $388 per lawn mower in production costs. How many lawn mowers were manufactured if the average cost of production is $660?
Answer:
128 lawn mowers
Step-by-step explanation:
Given,
The overhead cost = $ 34,816,
The cost for one lawn mower = $ 388,
Let x be the number of lawn mowers manufactured,
So, the total cost of x lawn mowers = 34816 + 388x,
Now, if the average cost of x mowers = $ 660,
So, the total cost = 660x
[tex]\implies 660x = 34816 + 388x[/tex]
[tex]660x - 388x = 34816[/tex]
[tex]272x= 34816[/tex]
[tex]\implies x = \frac{34816}{272}=128[/tex]
Hence, 128 lawn mowers were manufactured.
In how many different ways can five elements be selected in order from a set with three elements when repetition is allowed?
There are 243 ways to select five elements in order from a set of three elements with repetition allowed.
When selecting five elements in order from a set with three elements and repetition is allowed, each selection can include any of the three elements, repeated as necessary. Here's the breakdown:
1. For the first position, there are 3 choices.
2. For the second position, there are also 3 choices, as repetition is allowed.
3. Similarly, for the third, fourth, and fifth positions, there are 3 choices each.
To find the total number of ways, we multiply the number of choices for each position:
3 choices for the first position × 3 choices for the second position × 3 choices for the third position × 3 choices for the fourth position × 3 choices for the fifth position = [tex]\(3^5 = 243\)[/tex] ways.
Therefore, there are 243 different ways to select five elements in order from a set with three elements when repetition is allowed.
The correct naswer is 21.
The number of ways to select five elements in order from a set with three elements when repetition is allowed can be represented in LaTeX as:
[tex]\binom{5+3-1}{5} = \binom{7}{5} = \frac{7!}{5!(7-5)!} = \frac{7!}{5!2!} = 21[/tex]
Explanation:
- When repetition is allowed, the problem can be treated as finding the number of ways to arrange 5 objects with 3 distinct types.
- This can be solved using the combination formula, where we choose 5 positions out of 7 (5 elements + 3 distinct types - 1).
- The binomial coefficient [tex]\binom{n}{r}[/tex] represents the number of ways to choose [tex]$r$[/tex] items from a set of [tex]$n$[/tex] items.
- In this case, we are choosing 5 positions (elements) from a set of 7 positions (5 elements + 3 distinct types - 1).
- The binomial coefficient can be expanded using factorials: [tex]\binom{n}{r} = \frac{n!}{r!(n-r)!}[/tex]
- Substituting [tex]n = 7$ and $r = 5[/tex], we get[tex]\binom{7}{5} = \frac{7!}{5!(7-5)!} = \frac{7!}{5!2!} = 21[/tex]
Therefore, there are 21 different ways to select five elements in order from a set with three elements when repetition is allowed.
Sherita’s club is selling grapefruit to raise money. For every box they sell, they get $1.35 profit. They have sold 84 boxes already. How many more boxes must they sell to raise 270 dollars
HELP!!!!!!! I GIVE BRAINLEST AND THANKS!!!!!! + 5 POINTS!!!!
The volume of oil in four different containers is shown below: container
a.5.25 milliliters container
b.5.29 milliliters container
c.5.27 milliliters container
d.5.23 milliliters sue has a measuring cup that can measure to the nearest tenth of a milliliter. if sue measures the oil in each container, the least amount of oil would measure ____ milliliters.
Answer:
5.2
Step-by-step explanation:
14÷420 long division
According to the text, which is not a main component of drawing?
Perspective
Vanishing point
Horizon
For an angle θ with the point (–20, –21) on its terminating side, what is the value of cosine?
Answer:
-20/29
Step-by-step explanation:
eight times the sum of a and b
The question concerns a basic algebraic expression 'eight times the sum of a and b' which is represented as 8(a + b). The expression emphasizes the operation order: sum first, then multiply, which results in a value eight times greater than the original sum.
Explanation:The question 'eight times the sum of a and b' is a mathematical expression that can be represented as 8(a + b). This expression means you first add the numbers 'a' and 'b' and then multiply their sum by eight. The result you get after the multiplication will be eight times greater than the original sum of 'a' and 'b'. For instance, if 'a' is 2 and 'b' is 3, their sum is 5, and when this is multiplied by eight, it becomes 40, which is the desired expression's value.
To understand this concept further, we can refer to the exponentiation rule mentioned, which states that (xa)b = xa.b. Although this is a different type of operation—exponentiation—it demonstrates a similar principle of first performing the operation inside the parentheses and then applying the outside operation.
Finally, when performing algebraic operations, it's essential to remember that whatever you do to one side of the equation, you should do to the other side to maintain balance. This is a fundamental principle in algebra that helps to solve equations, such as the example provided showing how to isolate 'a' by subtracting 'x' from both sides of the equation a b.
A circle of radius 1 centered at (4, 0) is rotated about the y-axis.
1) Draw a picture of the three-dimensional shape that is produced when the circle is rotated about the y-axis.
2) In two or more complete sentences, describe the three-dimensional shape.
please help and thank you.
A gas station is 12 kilometers away. How far is the gas station in miles? Use the following conversion: 1 mile is 1.6 kilometers.
If h(x)=6-x, what is the value of (h o h)(10)
1. Explain a method of determining the correct degree and classification of a polynomial.
2. Why is the polynomial, 4x^2y + 5xy classified as a 3rd degree binomial?
what is the value of the fourth term in a geometric sequence for which a1=15 and r=1/3
express your answer as a fraction
Answer: The required fourth term in the given geometric sequence is [tex]\dfrac{5}{9}.[/tex]
Step-by-step explanation: We are given to find the fourth term of a geometric sequence with the following first term and common ratio :
[tex]a=15,~~r=\dfrac{1}{3}.[/tex]
We know that
the nth term of a geometric sequence with first term a and common ratio r is given by
[tex]a_n=ar^{n-1}.[/tex]
Therefore, the forth term of the given geometric sequence is
[tex]a_4=ar^{4-1}=ar^3=15\times\left(\dfrac{1}{3}\right)^3=15\times\dfrac{1}{27}=\dfrac{5}{9}.[/tex]
Thus, the required fourth term in the given geometric sequence is [tex]\dfrac{5}{9}.[/tex]
Find the lateral area for the prism. L.A. =
Find the total area for the prism. T.A. =
Answer:
Part 1) [tex]LA=(80+16\sqrt{13})\ in^{2}[/tex]
Part 2) [tex]TA=(104+16\sqrt{13})\ in^{2}[/tex]
Step-by-step explanation:
Part 1) Find the lateral area of the prism
we know that
The lateral area of the prism is equal to
[tex]LA=Ph[/tex]
where
P is the perimeter of the base
h is the height of the prism
Applying the Pythagoras Theorem
Find the hypotenuse of the triangle
[tex]c^{2}=4^{2}+6^{2}\\ \\c^{2}=52\\ \\c=2\sqrt{13}\ in[/tex]
Find the perimeter of triangle
[tex]P=4+6+2\sqrt{13}=(10+2\sqrt{13})\ in[/tex]
Find the lateral area
[tex]LA=Ph[/tex]
we have
[tex]P=(10+2\sqrt{13})\ in[/tex]
[tex]h=8\ in[/tex]
substitutes
[tex]LA=(10+2\sqrt{13})*8=(80+16\sqrt{13})\ in^{2}[/tex]
Part 2) Find the total area of the prism
we know that
The total area of the prism is equal to
[tex]TA=LA+2B[/tex]
where
LA is the lateral area of the prism
B is the area of the base of the prism
Find the area of the base B
The area of the base is equal to the area of the triangle
[tex]B=\frac{1}{2}bh[/tex]
substitute
[tex]B=\frac{1}{2}(6)(4)=12\ in^{2}[/tex]
Find the total area of the prism
[tex]TA=LA+2B[/tex]
we have
[tex]B=12\ in^{2}[/tex]
[tex]LA=(80+16\sqrt{13})\ in^{2}[/tex]
substitute
[tex]TA=(80+16\sqrt{13})+2(12)=(104+16\sqrt{13})\ in^{2}[/tex]
Factor the polynomial.
4x7+32x+5-24x^4
A. 4x^4(x^3+8x-6)
B. 2x^4(2x^3+16x-12)
C. 2x^4(x3+8x-6)
D. 4x^4(2x^3+16x-12)
Perform Gauss-Jordan elimination on the augmented matrix shown.
Could someone please help me figure this out.
What numbesr can be multiplied and added to equal the same number?
2
2x2 = 4
2+2 =4
0
0x0=0
0+0=0
Assume that month is an int variable whose value is 1 or 2 or 3 or 5 ... or 11 or 12. write an expression whose value is "jan" or "feb or "mar" or "apr" or "may" or "jun" or "jul" or "aug" or "sep" or "oct" or "nov" or "dec" based on the value of month. (so, if the value of month were 4 then the value of the expression would be "apr".).
The expression that satisfies the given condition is using if-else conditional statements to check the value of the variable 'month' and assign the corresponding month name.
Explanation:The expression that satisfies the given condition is:
if (month == 1) { answer = "jan"; } else if (month == 2) { answer = "feb"; } else if (month == 3) { answer = "mar"; } // ... continue this pattern for the remaining months
This code uses conditional statements (if-else) to check the value of the variable 'month' and assigns the corresponding month name to the variable 'answer'.
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the answer to number 3
Is the square root of 12 - 2 rational or irrational
Find the probability of drawing a king from a standard deck of cards and then drawing a queen after the first card is replaced in the deck. None of the above 1/26 1/13 1/2704 2/13
52 cards in a deck
4 kings
4 queens
King = 4/52 reduces to 1/13
Queen = 4/52 reduces to 1/13
1/13 *1/13 = 1/169
so none of the above