Zeros of a Polynomial Function
An important consequence of the Factor Theorem is that finding the zeros of a
polynomial is really the same thing as factoring it into linear factors. In this section we
will study more methods that help us find the real zeros of a polynomial, and thereby
factor the polynomial.
Rational Zeros of Polynomials:
The next theorem gives a method to determine all possible candidates for rational zeros
of a polynomial function with integer coefficients.
Rational Zeros Theorem:
If the polynomial ( ) 1
1 1 ... n n P x ax a x ax a n n
− = + ++ − + 0 has integer
coefficients, then every rational zero of P is of the form
p
q
where p is a factor of the constant coefficient 0 a
and q is a factor of the leading coefficient n a
Example 1: List all possible rational zeros given by the Rational Zeros Theorem of
P(x) = 6x
4
+ 7x
3
- 4 (but don’t check to see which actually are zeros) .
Solution:
Step 1: First we find all possible values of p, which are all the factors
of . Thus, p can be ±1, ±2, or ±4. 0 a = 4
Step 2: Next we find all possible values of q, which are all the factors
of 6. Thus, q can be ±1, ±2, ±3, or ±6. n a =
Step 3: Now we find the possible values of p
q by making combinations
of the values we found in Step 1 and Step 2. Thus, p
q will be of
the form factors of 4
factors of 6 . The possible p
q are
12412412412
, , , , , , , , , , ,
11122233366
± ± ± ± ± ± ± ± ± ± ±±
4
6
A student did six of ten problems correctly.
a. What is the ratio of the number right to the number wrong?
b. For every two answers that were wrong, how many answers were right?
5 feet.
The ratio is 3 : 2 and for every two incorrect problems he did three problems correctly.
Ratio, in mathematics, is a term used to compare two or more numbers. It is used to indicate the size of one quantity relative to another. In a report, two quantities are compared by division. Here the dividend is called the "prepayment" and the divisor is called the "consequence".For example, in a group of 30 people, 17 of them like to walk in the morning and 13 of them like to ride a bike. We use the ratio formula while comparing the relationship between two numbers or quantities. The general form of representing the ratio between two quantities, for example "a" and "b", is a:b, read as "a is b".i) Total number of problems =10
Correct problems = 6
Incorrect problems = 4
Ratio = Correct / incorrect
Ratio = 6/4
Ratio = 3/2
= 3 : 2
ii) The ratio of the number of correct problems to the number of bad problems, i.e. 3:2, represents that for every three problems the student can correct, both problems are wrong.
Hence, the ratio is 3 : 2 and for every two incorrect problems he did three problems correctly.
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24) The senior classes at High School A and High School B planned separate trips to Yellowstone National Park. The senior class at High School A rented and filled 13 vans and 2 buses with 230 students. High School B rented and filled 11 vans and 8 buses with 346 students. Every van had the same number of students in it as did the buses. How many students can a van carry? How many students can a bus carry?
The van can carry 13 students and the bus can carry 53 students.
A van can carry = 13
A bus can carry = 53
The equations will be derived from the conditions given in the question:
Let x be the no. of students in van
Let y be the no. of students in bus
13x + 9y = 646……………(1)
6x + 3y = 237………………(2)
Multiplying (2) with 3
13x + 9y = 646……………(3)
18x + 9y = 711 …………...(4)
Subtracting equation (3) from (4)
18x + 9y = 711
- (13x + 9y = 646)
We get:
5y = 65
y = 13
Now, plug the value of y into one of the preceding equations:
13x + 9(13) = 646
Solving this we get
x = 53
Hence, The van can carry 13 students and the bus can carry 53 students
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A car traveled 200 miles in 4 hours.What was the car´s average rate of speed in miles per hour
Answer:
50 miles per hour
Step-by-step explanation:
You want to put the total amount over the hours.
A figure has two pairs of parallel sides and four right angles. The figure is translated four units down. how many parallel sides does the image have?
Additionally, the picture would contain two parallel side pairs.
What are parallel lines?Lines that are parallel to one another on a plane do not intersect or meet at any point. They are always equidistant from one another and parallel. Non-intersecting lines are parallel lines. Parallel lines can also be said to meet at infinity.
How many sides of the picture are parallel?The parameters are as follows:
two parallel side pairsFour equal angles.4 units less translationA stiff transformation is the translation transformation.
This indicates that it maintains the orientation that is changing.
Two sets of parallel sides comprise the initial figure.
There would be two sets of parallel sides on the picture as a result.
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Find the sum. 69,348 + 76,290
Answer:
The answer you will be looking for is 145638!
If you need help how to i will explain!
solve and show your work.
3(2x+3) -4x = -3
A TV show had 3.6 x 10^4 viewers in the first week and 4.1 x 10^4 viewers in the second week. Determine the average number of viewers over the two weeks and write the final answer in scientific notation.
really need help pls pls
A salesclerk is creating a display of 16 polo shirts and 80 team shirts. Only one type of shirt can be in each row. What is the maximum number of shirts in each row so that all the rows are equal in length?
4
5
8
16
There may be a maximum of 16 shirts in each row to ensure that all the rows have the same length. Option D
What is the maximum number of shirts in each row so that all the rows are equal in length?The same elements at play
There are a total of sixteen polo shirts on exhibit.
80 jerseys from various teams are now on exhibit.
Each row is limited to just one kind of polo shirt.
The following are the factors that make up 80:
1, 2, 4, 5, 8, 10, 16, 20, and 80
The factors of 16 are as follows: 1, 2, 4, 8, 16.
The number 16 is the most frequent factor between the numbers 80 and 16.
Therefore, the greatest number of shirts that may be placed in each row such that the length of all of the rows is the same is 16.
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If the area of square 2 is 225units^2 and the perimeter of square 1bis 100. What is the area of square 3
Write the equation of a circle for which the endpoints of a diameter are (-0,0) and (4,-6)
The above statement describes the equation of a circle whose endpoints of a diameter are (-0,0) and (4,-6):
(x + 2)² + (y + 3)² = 3.6
How do we calculate the formula of a circle?The simple form equation of a circle is: (x – x1)² + (y – y1)²= r², where (x, y) are arbitrary coordinates upon that circumference of the circle, r seems to be the radius of the circle. and (x1, y1) are indeed the coordinates of the circle's center.
According to the given data:Given points are:
(-0,0) and (4,-6)
Find that diameter's length using the locations of the locations as well as the distance formula.
d = √((x₂ - x₁)² + (y₂ - y₁)²)
= √((4 - (-0))² + (-6 - 0)²)
= √((4 )² + (-6 )²)
= √(16 + 36)
= √(52)
= 7.2
Radius will be r/2
7.2/2
= 3.6
To obtain the coordinates of the center, use the interest in obtaining and the midpoint formula.
[tex]C=\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}[/tex]
C = (-0 + 4)/2 , (-6 + 0)/2
C = -2 , -3
To write this circle equation, use the radius and the center coordinates.
(x - h)² + (y - h)² = r²
(x - (-2))² + (y - (-3))² = 3.6
(x + 2)² + (y + 3)² = 3.6
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find the length of the side labled x
now, the picture looks a bit misleading, the angle on the bottom-right corner doesn't look at all like a 20° angle, so let me assume the 20° angle is the angle up above, like you see in the picture below, because that looks like a 20° one, and the number there, is not very clear which one is pointing to.
[tex]\cos(64^o )=\cfrac{\stackrel{adjacent}{h}}{\underset{hypotenuse}{18}}\implies 18\cos(64^o)=h \\\\[-0.35em] ~\dotfill\\\\ \cos(20^o )=\cfrac{\stackrel{adjacent}{h}}{\underset{hypotenuse}{x}}\implies x=\cfrac{h}{\cos(20^o )}\implies x=\cfrac{18\cos(64^o)}{\cos(20^o )}\implies x\approx 8.4[/tex]
Make sure your calculator is in Degree mode.
can somebody help me
The Initial amount is 14.76 and the decay factor is 1.4 while the percentage rate of change is 40%.
How to express exponential Functions?We are given the function as;
y = 1.4^(t + 8)
Now, the general formula for an exponential decay function is;
f(x) = abˣ
where;
a = initial amount
b = decay factor
x = number of time intervals
Now, using laws of exponents, our initial function can be written as;
y = 1.4⁸ × 1.4^(t)
Now, 1.4⁸ = 14.76
Thus, we now have the expression as;
y = 14.76(1.4^(t))
Comparing with the general equation of exponential decay gives us;
14.76 = initial amount
1.4 = decay factor
t = number of time intervals
If decay factor is 1.4, then this denotes 140% which means that the percentage rate of change is 40%
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what is the value of 5^5/5^2=?
By using the exponential value calculation, the value of 5^5/5^2 is 125.
How do we calculate an exponential value?The multiplier is represented by the base B, which is written as "B x," and the exponent "x," which indicates how many times the base is multiplied. When "8" is the base, "3" is the exponent, and the entire equation is the power, the result is 8X8X8=512.
It can be expressed as -
5⁵ / 5² = 5×5×5×5×5 / 5×5
= 5 × 5 × 5
= 5³
= 125
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Max worked 13 hours last week at the grocery store and earned 94.25. If he only worked 5 hours this week, how much money did he earn?
Answer:
36.25
Step-by-step explanation:
You need to find out how much he makes an hour. To do that you divide how much he made by 13 hours. Which was 7.25. Than multiply 7.25 by 5. And you get your answer which is 36.25.
The same goes for any money scenario when you are trying to find out like how much an individual 'peach' is in the whole bag. Or take it for what you will it was an example.
In a 36% sale, the price of a phone reduced by $135.
Find the original price of the phone.
Answer:
$210.94
Step-by-step explanation:
Can I have brainliest? It would help me out, if not thanks anyways! Hope this helped and have a nice day :)
Answer:
The original price was $375.
Step-by-step explanation:
The original price of the phone is x.
36% of x is $135.
0.36x = 135
x = 135/0.36
x = 375
Answer: The original price was $375.
Find m∠ABC and m∠CBD
5. m∠ABC = 64°
m∠CBD = 26°
6. Using the linear pair postulate,
m∠ABC = 97°
m∠CBD = 83°
7. Using the linear pair postulate,
m∠ABC = 112.5°
m∠CBD = 67.5°
How to Apply the Linear Angles Postulate?Two angles on a straight line form a linear pair which have a sum of 180 degrees based on the linear pair postulate.
5. (11x - 2) + (5x - 4) = 90 [complementary angles]
11x - 2 + 5x - 4 = 90
16x - 6 = 90
16x = 90 + 6
16x = 96
x = 6
m∠ABC = 11x - 2 = 11(6) - 2 = 64°
m∠CBD = 5x - 4 = 5(6) - 4 = 26°
6. 8x + 9 + 6x + 17 = 180 [linear pair postulate]
14x + 26 = 180
14x = 180 - 26
14x = 154
x = 11
m∠ABC = 8x + 9 = 8(11) + 9 = 97°
m∠CBD = 6x + 17 = 6(11) + 17 = 83°
7. 14x - 6.5 + 7x + 8 = 180 [linear pair postulate]
21x + 1.5 = 180
21x = 180 - 1.5
21x = 178.5
x = 8.5
m∠ABC = 14x - 6.5 = 14(8.5) - 6.5 = 112.5°
m∠CBD = 7x + 8 = 7(8.5) + 8 = 67.5°
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Chapter 15 Questions 4) In the diagram below, ABCD is a rectangle and triangle DEF is formed by joining the midpoints of AB and BC. a) determine the area of rectangle ABCD in terms of x and y, b) If the shaded area of triangle DEF is 15m², determine the value of xy. c) Hence, determine the area of triangle EBF
a) The area of rectangle ABCD in terms of x and y = 4xy.
b) xy = 10
c) area of triangle EBF = 5 m².
What is defined as the rectangle?A rectangle is a covered two-dimensional shape with four sides, four corners, and four right angles (90°). A rectangle's opposite sides are equal as well as parallel. Because a rectangle is a 2-D shape, it has two dimensions: length and width. The length of the rectangle is the longer side, and the width is really the shorter side.Part A) The area of rectangle ABCD:
See the diagram;
The length of rectangle is 2x.
The width of breadth is 2y.
Area = length×breadth
Area = 2x × 2y
Area of ABCD = 4xy m².
Part b) value of xy.
The area of the triangle DEF is 15m².
area ΔDEF = area ABCD - area ΔADE - area ΔBEF - area ΔDFC.
Put the values;
area ΔDEF = 4xy - 2xy/2 - 2xy/2 - xy/2
area ΔDEF = 3xy
equate;
ΔDEF = 3xy and DEF = 15m².
3xy/2 = 15
xy = 10
Thus, the value of xy = 10.
Part c) area of triangle EBF:
area ΔEBF = xy/2
Put the value of xy = 10.
area ΔEBF = 10/2 = 5
Thus, the area of the triangle EBF 10m².
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Hi! I really need help with the answer to this question!
The room has a 68 by 33 square foot space and perimeter.
based on the information provided:
a) The room's area is 8 x 8.5,
=68 square feet.
and the room's circumference is
2(8+8.5) = 33 ft
b). Yes, the baseboard and carpet will have the same area and perimeter.
What is a rectangle's perimeter?
Considering that the polygon's perimeter is equal to the sum of its sides. As a result, the perimeter (P) of a rectangle is given by P = the total of its four sides. P = a + b + a + b (Opposite sides of rectangle are equal)
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gaussian classes with distinct means, equal class prior probabilities, and common covariance matrix that is not necessarily a diagonal matrix.
The equal class prior probabilities, gaussian classes with unique means, and a common covariance matrix that isn't always diagonal. The particular choice for ∑ where d is 2 is µ[tex]_{2}[/tex]-µ[tex]_{1}[/tex].
What is a gaussian function?A Gaussian function in mathematics is a function with the base form and parametric extension for all real constants a, b, and c that are not zero. This function is frequently referred to as a Gaussian. It bears the name of the mathematician Carl Friedrich Gauss. A Gaussian's graph has the recognizable symmetric "bell curve" shape. The parameters a and b determine the height and location of the curve's peak, respectively, while the parameter c, also known as the standard deviation or Gaussian RMS width, determines the width of the "bell." Gaussian functions are frequently used in statistics to characterize the normal distributions, in signal processing to build Gaussian filters, and in image processing where two-dimensional Gaussians are utilized for Gaussian blurring.
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find the largest value of $n$ such that $5x^2 nx 48$ can be factored as the product of two linear factors with integer coefficients.
If we can write
[tex]5x^2 + nx + 48 = (ax + b) (cx + d)[/tex]
then expanding the right side gives
[tex]5x^2 + nx + 48 = acx^2 + (ad+bc)x + bd[/tex]
so that [tex]ac=5[/tex], [tex]ad+bc=n[/tex], and [tex]bd=48[/tex], where [tex]a,b,c,d[/tex] are integers. We want to maximize [tex]ad+bc[/tex].
5 is prime, so [tex](a,c)[/tex] can be either (1, 5) or (5, 1). Let [tex]a=5[/tex] and [tex]b=1[/tex]. Then maximizing
[tex]ad+bc=5d + c[/tex]
is just a matter of picking the largest possible value for [tex]d[/tex], which is 48. Then [tex]\max\{ad+bc\}=5\times48+1\times1=\boxed{241}[/tex].
Solve equation.
k+3=14
To solve the equation, subtract 3 from both sides. The solution is k = 11.
A linear equation is an equation of which the highest degree of its variables is 1.
Steps to solve linear equations:
Simplify the expressions. If there are parentheses, remove them by multiplying the corresponding terms.Combine the same terms.Isolate the variable on one side by using subtraction of addition.Use division or multiplication to find the value of the variable.Recall that if we subtract or add the same number to both sides of an equation, it does not change the equality.
The given problem:
Solve k + 3 = 14.
The variable is k. Subtract 3 from both sides:
k + 3 - 3 = 14 - 3
k = 11 (solved)
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which paired number is a solution to the equation y=-9x+4
(10,-86)
(-4,-58)
(6,-41)
(-6,57)
Answer:
(10,-86)
Step-by-step explanation:
Plug in the x values to see if they equal to the y value.
-9(10)+4 = -86.
x = 10
y = -86
Mount Everest has an elevation of about 8850 meters.
Any location above 8000 meters is in the death zone,
which is the elevation at which there is not enough
oxygen to sustain human life. Write an inequality
that represents the possible elevations of climbers
in the death zone on Mount Everest.
The inequality that represents the possible elevations of climbers in the death zone on Mount Everest is 8000 < x ≤ 8850
How to write an inequality that represents the possible elevations of climbers in the death zone on Mount Everest.The given parameter is'
Height = About 8850 meters
Death zone = Location above 8000 meters
Let the height be represented with x
So, the inequality that represents x is
Death zone < x ≤ Height
This gives
8000 < x ≤ 8850
Hence, the inequality that represents the possible elevations of climbers in the death zone on Mount Everest is 8000 < x ≤ 8850
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4. Find the base of a triangle if its height
is 2 3/4 cm and its area is 12 5/8cm². Write
your answer as a mixed number.
The base of the triangle is 9 2/11 cm.
Given the height of the triangle is = 2 3/4 cm
area of the triangle is = 12 5/8 cm²
we know the area of the triangle is = 1/2 × b × h
but base = ?
therefore, 12 5/8 cm² = 1/2 × b × 2 3/4 cm
convert improper fraction to proper fraction.
101/8 = 1/2 × b × 11/4
101/8 = 11b/8
like terms gets cancelled
101 = 11b
b = 101/11
b = 9 2/11 cm
Hence we get the base of the triangle as 9 2/11 cm.
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does someone mind helping me with this? Thank you!
Answer:
196m²
Step-by-step explanation:
area is w multiplied by h so 14x14
Answer:
196 [tex]m^{2}[/tex]
Step-by-step explanation:
The area of a rectangle is length times width. A square is a type of rectangle that has four equal sides, so if we know one side we know all four.
A = l x w
A = 14m x 14m
A = 196[tex]m^{2}[/tex]
(6d+5)−(2−3d) = what is the answer
[tex]{ \red{ \bold{9d}}} \: + \: { \red{ \bold{3}}} [/tex]
Step-by-step explanation:
[tex]{ \blue{ \tt{(6d + 5)}}} - { \blue{ \tt{(2 - 3d)}}}[/tex]
[tex]{ \blue{ \tt{6d + 5 - 2 + 3d}}}[/tex]
[tex] = { \blue{ \tt{9d + 3}}}[/tex]
The square of T varies directly with the cube of a and inversely with the square of d; T = 2 when a = 2 and d = 4
The general formula that describes the variation is T² = 18a³/d²
Direct variation: When a variable x changes directly with another variable y, it is written in this form;x ∝ y.
This can then be written as;
x = ky
Where;
k = constant rate of change.
Inverse: When one variable x changes in the opposite direction to another variable y, it is written in this form; x ∝ 1 / y This can then be written as; x = k (1/y) Where; k = constant rate or variationCombined variation: When a variable x varies directly with variable y and inversely with variable z, we write it in this form;x ∝ (y/z)
This can then be written as;
x = k (y/z)
Where;
k = constant rate or variation
The square of T varies directly with the cube of a and inversely with the square of d.
Note that
square of T = T²
cube of a = a³
square of d = d²
Therefore, we can write;
T² ∝ a³/d²
T² = k (a³/d²)
Since;
T = 4 when a = 2 and d = 3
We can find the constant of proportionality k, by substituting the values of T=4, a = 2 and d = 3 into equation (i) and solve as follows;
(4)² = k (2³/3²)
16 = k (8/9)
8k = 16 x 9
8k = 144
k = 144/8
k = 18
Therefore, The overall formula for describing the variation is
T² = 18 a³/d²
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=
REAL NUMBERS
Simplifying a ratio of whole numbers: Problem type 1
Write the ratio as a fraction in simplest form, with whole numbers in the numerator and denominator.
35 to 30
The simplest form of the ratio 35:30 is 7/6.
Given, the ratio is 35:30
By comparing two amounts of the same units, the ratio can be used to calculate how much of one item is included in the other. There are two different categories for ratios. The first is a part to part ratio, whereas the second is a part to whole ratio. The relationship between two separate entities or organizations is shown by the part-to-part ratio.
We can refer to a fraction as being in its simplest form when it cannot be further spelled out or reduced. Divide the denominator and numerator by their respective least common multiples to reduce a fraction.
divide the numbers by 5
35/30
= 7/6
therefore the simplest fraction is 7/6.
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Eli and Ana have the same number of dumplings for dinner. There are d dumplings in each container. Which equation represents the situation?
The equation that represents the equal distribution of ' d ' dumplings among Eli and Ana is -
x = nd/2.
What is unitary method?
Unitary method is used to find the value of a single unit and then multiply the value of a single unit to the number of units to get the necessary value.
Given is a situation in which Eli and Ana are having dinner. They both have same number of dumplings to eat and each dumpling container contains 'd' dumplings.
Let's assume that there are 'n' containers of dumplings and both Eli and Ana got 'x' number of dumplings. Now -
In 1 container we have - 'd' dumplings
In 'n' containers, we will have - n × d dumplings.
Both Ana and Eli got same number of dumplings = x.
Therefore -
x + x = n × d
2x = nd
x = nd/2
Therefore, the equation that represents the situation is -
x = nd/2
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at an inter-species dance party, each cat danced with exactly three dogs and each dog danced with exactly two cats. no one danced with anyone of their own species, and twelve cats attended the party. how many dogs attended the party? (a) 8 (b) 12 (c) 16 (d) 18 (e) 24
The correct option is (d) 18.
The number if dogs attended the inter-species dance party is 18.
What is combination?A combination is an arithmetical technique for determining the number of different arrangements in a set of items in which the order of a selection is irrelevant. You can choose the items in just about any order in combinations.
Permutations and combinations are often confused. In permutations, however, the sequence of the selected items is critical. For instance, the arrangements ab as well as ba are equal in combinations (regarded like one arrangement), but different in permutations.Combinations are analyzed in combinatorics, but they are also used in other fields such as mathematics and finance.Now, as per the given question.
If every cat danced with a 3 dogs, the total would be;
3 x 12 = 36 cat and dogs pairs.
Even so, every dogs just danced with 2 cat, implying that there had to be a mix-up. Then,
Number of dogs = 36/2 = 18
Therefore, the total number of dogs present in the inter-species dance party are 18.
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