the polynomial p(x)=2x^3+17x^2+41x+30 has a known factor of (x+5) rewrite p(x) as a product of linear functionp(x)=
Answers
Answer:
Rewriting p(x) as a product of linear functions, we have;
[tex]p(x)=(2x+3)(x+5)(x+2)[/tex]Explanation:
We want to write the given polynomial p(x) as a product of linear functions.
[tex]p(x)=2x^3+17x^2+41x+30[/tex]To write it as a product of linear functions we have to find the other factors;
let us divide the given polynomial by the given factor;
[tex]\begin{gathered} \text{ }2x^2+7x+6 \\ (x+5)\sqrt[]{2x^3+17x^2+41x+30} \\ \text{ - (}2x^3+10x^2) \\ \text{ }7x^2+41x+30 \\ \text{ }-(7x^2+35x) \\ \text{ }6x+30 \\ \text{ - (}6x+30) \\ \text{ 0} \end{gathered}[/tex]So, the division gives;
[tex]p(x)=2x^3+17x^2+41x+30=(x+5)(2x^2+7x+6)[/tex]next, we need to find the factors of the quadratic function;
[tex]\begin{gathered} 2x^2+7x+6 \\ 2x^2+4x+3x+6 \\ 2x(x+2)+3(x+2) \\ (2x+3)(x+2)_{} \end{gathered}[/tex]Substituting the factors of the quadratic function, we have;
[tex]\begin{gathered} p(x)=2x^3+17x^2+41x+30=(x+5)(2x^2+7x+6) \\ p(x)=(x+5)(2x+3)(x+2)_{} \\ p(x)=(2x+3)(x+5)(x+2) \end{gathered}[/tex]Therefore, rewriting p(x) as a product of linear functions, we have;
[tex]p(x)=(2x+3)(x+5)(x+2)[/tex]Related Questions
-2. The sum of two cubes can be factored by using the formula o’ + b3 (a + b)(c? ab + b?).(a) Verify the formula by multiplying the right side of the equation.(b) Factor the expression 8x2 + 27.(C) One of the factors of q? - bºis a - b. Find a quadratic factor of q? - bº. Show your work.(d) Factor the expression x - 1.
Answers
Given that the sum of two cubes can be factored by using the formula
[tex]a^3+b^3=(a+b)(a^2-ab+b^2)[/tex]a) To verify the formula by multiplying the right side equation
[tex]\begin{gathered} (a+b)(a^2-ab+b^2) \\ =a(a^2-ab+b^2)+b(a^2-ab+b^2) \\ =a^3-a^2b+ab^2+a^2b-ab^2+b^3 \\ \text{Collect like terms} \\ =a^3-a^2b+a^2b+ab^2-ab^2+b^3 \\ \text{Simplify} \\ =a^3+b^3 \end{gathered}[/tex]Hence,
[tex](a+b)(a^2-ab+b^2)=a^3+b^3[/tex]b) To factor
[tex]8x^3+27[/tex]Using the sum of two cubes formula, i.e
[tex]a^3+b^3=(a+b)(a^2-ab+b^2)[/tex]Factorizing the expression gives
[tex]\begin{gathered} (2x)^3+(3)^3=(2x+3)((2x)^2-(2x)(3)+(3)^2)_{} \\ (2x)^3+(3)^3=(2x+3)(4x^2-6x+9) \end{gathered}[/tex]Hence, the answer is
[tex](2x+3)(4x^2-6x+9)[/tex]c) Given that one of the factors of a³ - b³ is a- b, the quadratic factor of a³ - b³ can be deduced by applying the differences of cubes formula below
[tex]a^3-b^3=(a-b)(a^2+ab+b^2)^{}_{}[/tex]Expanding the right side equations
[tex]\begin{gathered} (a-b)(a^2+ab+b^2)^{}_{}=a(a^2+ab+b^2)-b(a^2+ab+b^2) \\ =a^3+a^2b+ab^2-a^2b-ab^2-b^3 \\ \text{Collect like terms} \\ =a^3+a^2b-a^2b+ab^2-ab^2-b^3 \\ \text{Simplify} \\ =a^3-b^3 \end{gathered}[/tex]Hence, the quadratic factor is
[tex]a^3-b^3=(a-b)(a^2+ab+b^2)[/tex]d) To factor the expression
[tex]x^3-1[/tex]By applying the differences of cubes formula
[tex]a^3-b^3=(a-b)(a^2+ab+b^2)[/tex]Factorizing the expression gives
[tex]\begin{gathered} (x)^3-(1)^3=(x-1)(x^2+(x)(1)+1^2)^{}_{} \\ x^3-1^3=(x-1)(x^2+x+1) \end{gathered}[/tex]Hence, the answer is
[tex](x-1)(x^2+x+1)[/tex]
The perfect squares between 54 and 102.
Answers
The perfect squares are between 54 and 102.
54 and 102 are taken,
7 × 7 = 49
8 × 8 = 64
9 × 9 = 81
10 × 10 = 100
11 × 11 = 121
Here, the perfect squares between 54 and 102 are 8, 9, 10.
So, answer is 8 × 8, 9 × 9, 10 × 10
Which line represents the best fit for the scatter plot data?
Answers
A line of best fit can be roughly determined by drawing a straight line on a scatter plot so that the number of points above and below the line is about equal and the line passes through as many points as possible.
When we look at the fits in B and C, we notice that the line is not following the behavior of the data accurately, so we can rule out these options.
When we look at option D, we can see that the line is above all the points, we can also notice that this line doesn't go through any point of the data, that's why option A, is the best fit for this set of data, since it passes through some points and there are points above and below the line.
The correct graph is A
brainstorm some real-world applications where integration could be helpful, then describe your examples and explain how integration fits in
Answers
Given
Integration
Find
some real-world applications where integration could be helpful
Explanation
In real life , integrations are used in vaarious fields such as engineering , where engineers use integrals to find the slope of the building
In physics , it is used in the centre of gravity .
In field of graphical representation , where three- dimensional models are demonstrated.
Final Answer
Hence , above are the some real world applications.
Complete the following tables 1 and 3 only.
Answers
Answer:
(a) D = 5/8 in
(b) A = 31.2 cm
(c) B = 8 ft
(d) C = 3 m
Explanation:
If two triangles are similar, their corresponding sides are proportional, so we can always use the following equation:
[tex]\frac{A}{B}=\frac{C}{D}[/tex]Therefore, for row (a), we can write the following equation:
[tex]\frac{5\frac{1}{2}}{1\frac{1}{4}}=\frac{2\frac{3}{4}}{D}[/tex]So, changing the mixed number by decimals and solving for D, we get:
[tex]\begin{gathered} \frac{5.5}{1.25}=\frac{2.75}{D} \\ 5.5D=2.75(1.25) \\ 5.5D=3.4375 \\ \frac{5.5D}{5.5}=\frac{3.4375}{5.5} \\ D=0.625 \end{gathered}[/tex]Then, for row (a), D = 0.625 = 5/8
In the same way, we can write and solve the following equation for row (b)
[tex]\begin{gathered} \frac{A}{23.4}=\frac{20.8}{15.6} \\ \frac{A}{23.4}\times23.4=\frac{20.8}{15.6}\times23.4 \\ A=31.2 \end{gathered}[/tex]For row (c), we get:
[tex]\begin{gathered} \frac{12}{B}=\frac{9}{6} \\ 12(6)=9(B) \\ 72=9B \\ \frac{72}{9}=\frac{9B}{9} \\ 8=B \end{gathered}[/tex]For row (d), we get:
[tex]\begin{gathered} \frac{4.5}{3.6}=\frac{C}{2.4} \\ \frac{4.5}{3.6}\times2.4=\frac{C}{2.4}\times2.4 \\ 3=C \end{gathered}[/tex]Therefore, the answers are:
(a) D = 5/8 in
(b) A = 31.2 cm
(c) B = 8 ft
(d) C = 3 m
Find a recursive rule for the nth term of the sequence.7, 28, 112, 448, ...
Answers
Solution
Given the sequence 7, 28, 112, 448, ...
The sequence is a Geometric sequence because it has a common ratio
[tex]Common\text{ ratio, r = }\frac{28}{7}=4[/tex]First term, a = 4
[tex]\begin{gathered} The\text{ nth term of a gp = ar}^{n-1} \\ Where\text{ n=number of terms} \\ a=\text{ first term } \\ r=common\text{ ratio} \end{gathered}[/tex][tex]T_n=7\text{ \lparen4\rparen}^{n-1}[/tex][tex]\begin{gathered} For\text{ recursive, } \\ T_n=r.T_{n-1} \\ T_n=4(T_{n-1}) \end{gathered}[/tex][tex]The\text{ recursive rule is 4\lparen T}_{n-1})[/tex]Which measurement is the best estimate for the volume of the figure? Roundeach measurement to the nearest whole number to get your estimate.A. 9 cubic metersB. 12 cubic metersC. 6 cubic metersD. 8 cubic meters
Answers
we have that
The volume of the rectangular prism is given by
[tex]V=L*W*H[/tex]where
L=3.5 m --------> 4 m
W=0.7 m -------> 1 m
H=2.2 m -------> 2 m
substitute
[tex]\begin{gathered} V=(4)(1)(2) \\ V=8\text{ m}^3 \end{gathered}[/tex]The answer is option DThe portable basketball hoop shown is made so that BA = AS = AK. The measure of < BAK is 128 degrees. Calculate m < BSK.
Answers
The measure of ∠BSK is 64 degrees, that is the value of m∠BSK is 64 degrees.
We are given BA = AS = AK.
∠BAK = 128 degrees
From the linear pair concept:
∠BAK + ∠SAK = 180 degrees
128 degrees + ∠SAK = 180 degrees
∠SAK = 180 degrees - 128 degrees
∠SAK = 52 degrees
From the angle sum property of a triangle in triangle ASK, we will get;
∠SAK + ∠ASK + ∠AKS = 180 degrees
52 degrees + ∠ASK + ∠AKS = 180 degrees
2 ∠ASK = 180 degrees - 52 degrees (Since, AS = AK)
∠ASK = 128/2 degrees
∠ASK = 64 degrees
Thus, the measure of ∠BSK is 64 degrees, that is the value of m∠BSK is 64 degrees.
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Solve the equation.56 = 13 + d
Answers
ANSWER
d = 43
EXPLANATION
We want to solve the equation for d.
The given equation is:
56 = 13 + d
To solve it, we collect like terms and simplify. That is:
56 - 13 = d
=> d = 43
That is the solution.
Determine the scale factor of the dilation. Write your answer as a fraction if necessary.
Answers
The scale factor of the dilation is 1.5
Here, we want to determine the scale factor of the dialtion
From what we can see, we have the smaller, being moved to the bigger and thus, we expect a scale factor higher than 1
We can work with any of the side lengths to determine the scale factor
Let us look at the base
For the smaller shape, we have a length from (1,-2) to (-2,-2)
For the bigger shape, we have a length from (1.5,-3) to (-3,-3)
Calculating these distances, we have a unit of 3 units in the smaller and a unit of 4.5 units in the bigger
So, we have the scale factor as;
[tex]\frac{4.5}{3}\text{ = 1.5}[/tex]What lengths are possible for pieces cut from the 27-cm piece of string? Select all that apply. A. 2 cm B. 7 cm C. 5 cm D. 3 cm E. 9 cm F. 27 cm G. 18 cm H. 1 cm
Answers
Answer:
D, E, F, and H
Step-by-step explanation:
D. 27/3=9
E. 27/9=3
F. 27/27=1
H. 27/1=27
Answer: D,E,F, and H
Step-by-step explanation:
1, 3, 9, and 27 are the multiples of 27 itself
It can’t be 2 because it’s not an even number
It can’t be 7 because the closest number that is a multiple of 7 is 28 and it’s one over
It can’t be 5 because it doesn’t end in 5 or 0
it can’t be 18 because 18 + 18 equals 36 and that is 9 over the original number
Find the radius of a circle on which a central angle measuring 5π/6 radians intercepts an arc on the circle with a length of 35π centimeters.A. 42 cmB. 37 cmC. 61 cmD. 44 cm
Answers
Solution:
the arc lenght s, of a circle of radius r, with central angle θ, is given by the following equation:
[tex]s\text{ = }r\theta[/tex]solving for r, we get the following equation:
[tex]r\text{ = }\frac{s}{\theta}[/tex]now, replacing in the previous equation the data given in the problem we get:
[tex]r\text{ = }\frac{s}{\theta}\text{ = }\frac{35\pi}{5\pi\text{ / 6}}\text{ = }\frac{(35\text{ }\pi)\text{ 6 }}{5\pi}\text{ = 42 }[/tex]so that, we can conclude that the correct answer is:
[tex]42\text{ cm}[/tex]Given:• 1 cm^3= 1 mL• 1 dm^3 = 1 L• 1L = 1,000 mLIf a health person's kidneys can filter 125 mL of blood per minute, then how long will it take for the kidneys to filter 4.5 L of blood?
Answers
Convert the following measurement 19 quarts to cups
Answers
76 cups
Explanation:Note that:
1 quart = 4 cups
Therefore:
19 quarts = 4 x 19 cups
19 quarts = 76 cups
Summer earns 15% commission onsales for a software company. If inone week she makes three sales.one at $375, one at $1.200, and oneat $900. how much did Summerearn that week?
Answers
She made
[tex]375+1200+900=2475[/tex]in sales. Then, she earned
[tex]0.15(2475)=371.25[/tex]$371.25 that week.
find the coordinates of the ordered pair where the maximum value occurs for equation P equals 5 x + 5y + 42 given these constrains
Answers
Given the equation:
[tex]P=5x+5y+42[/tex]Given the constraints:
[tex]\begin{gathered} -2x+4y\ge-4 \\ x\ge-8 \\ y\le9 \end{gathered}[/tex]Let's find the ordered pair where the maximum value occurs for P.
From the inequality of let's solve for x and y.
At x = -8:
[tex]\begin{gathered} -2x+4y=-4 \\ -2(-8)+4y=-4 \\ \\ 16+4y=-4 \\ \\ \text{Subtract 16 from both sides:} \\ 16-16+4y=-4-16 \\ 4y=-20 \\ \\ \text{Divide both sides by 4:} \\ \frac{4y}{4}=\frac{-20}{4} \\ \\ y=-5 \\ \\ \text{Thus, we have the points:} \\ (x,y)\Longrightarrow(8,-5) \end{gathered}[/tex]At y = 9:
[tex]\begin{gathered} -2x+4y=-4 \\ -2x+4(9)=-4 \\ -2x+36=-4 \\ \\ \text{Subtract 36 from both sides:} \\ -2x+36-36=-4-36 \\ -2x=-40 \\ \\ \text{Divide both sides by -2:} \\ \frac{-2x}{-2}=\frac{-40}{-2} \\ \\ x=20 \\ \\ \text{thus, we have the point:} \\ (x,y)\Longrightarrow(20,9) \end{gathered}[/tex]Input the values of x and y into the equation and solve for evaluate for P.
• (x, y) ==> (-8, -5):
P = 5x + 5y + 42
P = 5(-8) + 5(-5) + 42
P = -40 - 25 + 42
P = -23
• (x, y) ==> (20, 9):
P = 5x + 5y + 42
P = 5(20) + 5(9) + 42
P = 100 + 45 + 42
P = 187
We can see the maximum value of P is 187 at (20, 9)
The maximum value occurs at (20, 9)
• ANSWER:
(20, 9)
Consider the function.f(x) = x2 − 1, x ≥ 1
Answers
t Given
[tex]f(x)=\sqrt{x^2-1}[/tex]Find
inverse of f(x)
domain and range of function and its inverse
Explanation
Let y = f(x)
[tex]y=\sqrt{x^2-1}[/tex]replace all x with y and y with x
[tex]x=\sqrt{y^2-1}[/tex]now solve for y
[tex]\begin{gathered} \sqrt{y^2-1}=x \\ y^2-1=x^2 \\ y^2=x^2+1 \\ y=\pm\sqrt{x^2+1} \end{gathered}[/tex]so, the inverse is
[tex]\pm\sqrt{x^2+1}[/tex]domain of f(x) is
[tex]x\epsilon R:x\leq-1\text{ or x}\ge1[/tex]range of f(x) is
[tex]y\epsilon R:y\ge0[/tex]domain of inverse is
[tex]R[/tex]range of inverse function is
[tex]y\epsilon R:y\ge1[/tex]Find the slope from the tableA. 3B. 2C. -3D. -1
Answers
Use 2 of the given ordered pairs to find the slope of the function. Use the following formula:
[tex]m=\frac{y2-y1}{x2-x1}[/tex]Where y2 and y1 are the y coordinates of the ordered pairs and x2 and x1 are the x coordinates. Replace for the given values:
[tex]m=\frac{2-5}{-2-(-3)}=-\frac{3}{1}=-3[/tex]The slope of the function is -3.
Charnaie owns her own tutoring service. She charges new clients $20 for a placement test and then $10 per hour for every hour of tutoring.
Drag and drop the boxes to correctly complete the statements.
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
What is the independent variable in this real-world situation? Explain your reasoning.
The independent variable is the Response area because Response area.
What is the dependent variable in the real-world situation? Explain your reasoning.
The dependent variable is the Response area because Response area.
Answers
The correct equation for the given condition will be;
⇒ T = $20 + $10h
Where, T is the total cost and 'h' is the number of hours.
What is an expression?
Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
She charges new clients $20 for a placement test and $10 per hour for every hour of tutoring.
Now,
Let total cost of charge = T
And, Number of hours = h
So, We can formulate by the given condition as;
⇒ T = $20 + $10h
Thus, The correct equation for the given condition will be;
⇒ T = $20 + $10h
Where, T is the total cost and 'h' is the number of hours.
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what is 4 2/3 + 7/9 as a fraction
Answers
The calculation is
[tex]4\cdot\frac{2}{3}+\frac{7}{9}[/tex]First step is to solve the multiplication.
4 means that there are four wholes, if you express in in thirds
[tex]\begin{gathered} 1\text{whole}=\frac{3}{3} \\ 4\text{wholes}=\frac{3\cdot4}{3}=\frac{12}{3} \end{gathered}[/tex]Then the multiplication you have to do is
[tex]\frac{12}{3}\cdot\frac{2}{3}=\frac{8}{3}[/tex]Now that the multiplication is done add 7/9
[tex]\frac{8}{3}+\frac{7}{9}=\frac{31}{9}[/tex]What is the linear equation ( slope intercept equation ) for the line that passes through points (0,4) and (2,8) ?
Answers
The equation of a line in the slope intercept form is expressed as
y = mx + c
where
m represents slope
c represents y intercept
The formula for determining slope is expressed as
m = (y2 - y1)/(x2 - x1)
Consideing the given points,
x1 = 0, y1 = 4
x2 = 2, y2 = 8
m = (8 - 4)/(2 - 0) = 4/2
m = 2
We would find the y intercept, c by substituting m = 2, x = 0 and y = 4 into the slope intercept equation. It becomes
4 = 2 * 0 + c
c = 4
Substituting m = 2 and c = 4 into the slope intercept equation, it becomes
y = 2x + 4
The last option is correct
what is the distance between points A(3,12) and B(6,15)? round to the nearest whole number
Answers
The distance between two points is given by:
[tex]d(A,B)=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]Then, in our case, we have:
[tex]\begin{gathered} d(A,B)=\sqrt[]{(6-3)^2+(15-12)^2} \\ =\sqrt[]{(3)^2+(3)^2} \\ =\sqrt[]{9+9} \\ =\sqrt[]{18} \\ =4.2426 \end{gathered}[/tex]Therefore the distance between the points (rounded to the nearest whole number) is 4.
69 is _________% more than 60
Answers
To find the solution we can use the rule of three:
[tex]\begin{gathered} 60\rightarrow100 \\ 69\rightarrow x \end{gathered}[/tex]then:
[tex]\begin{gathered} x=\frac{69\cdot100}{60} \\ x=115 \end{gathered}[/tex]This means that 69 is 115% of 60.
Therefore 69 is 15% more than 60.
Need help with a precalc question
Answers
Given that
The equation is
[tex][/tex]write 37,000,010 numbers using words
Answers
thirty-seven million ten
Explanation
Step 1
count the number of digits after 37
[tex]\begin{gathered} 37000010 \\ 6\text{ digits, it meas} \\ by\text{ now, we have} \end{gathered}[/tex]thirty-seven million
Step 2
the remaining number is
[tex]\begin{gathered} 000010 \\ it\text{ is , ten} \\ 10 \end{gathered}[/tex]ten
Step 3
combine
thirty-seven million ten
I need help with geometry. I am supposed to solve for x in this diagram and assume lines marked with interior arrowheads are parallel :)
Answers
ANSWER:
40°
STEP-BY-STEP EXPLANATION:
We can make the following equality thanks to the properties of these angles:
[tex]\begin{gathered} 3x=120 \\ \text{ solving for x} \\ x=\frac{120}{3} \\ x=40\text{\degree} \end{gathered}[/tex]The value of x is 40°
solving absolute value inequalities[tex]8 - 9 |6x - 10| \ \textgreater \ - 82[/tex]please help with steps, I keep getting stuck on this one
Answers
For (1):
[tex]\begin{gathered} 6x-10<82 \\ \text{Add 10 to both sides:} \\ 6x<82+10 \\ 6x<92 \\ \text{Divide both sides by 6:} \\ x<\frac{92}{6} \\ x<\frac{46}{3} \\ \end{gathered}[/tex]For (2):
[tex]\begin{gathered} 6x-10>-82 \\ \text{Add 10 to both sides:} \\ 6x>-82+10 \\ 6x>-72 \\ \text{Divide both sides by 6:} \\ x>-\frac{72}{6} \\ x>-12 \end{gathered}[/tex]Therefore, the solution is:
[tex]-12Which of the following represents the factorization of the polynomial functiongraphed below? (Assume it has no constant factor.)O A. y - (x - 1)(x+5)OB. y - (x + 1)(x+5)O C. y - (x + 1)(x - 5)O D. y = (x - 1)(x-5)
Answers
The graph of the function intersect the x-axis at x = 1 and x = 5. So zeos of the function is,
[tex]x=1\text{ and x = 5}[/tex]So factorization quation of polynomial equation is,
[tex]y=(x-1)(x-5)[/tex]Answer: y = (x - 1)(x - 5)
dominic is making meatballs. he uses 3/4 cup of breadcrumbs for every 1 1/4 pounds of ground beef. how many cups of bradecrumbs does he need when he uses 1 3/4 pounds of ground beef?
Answers
The number of cups of breadcrumbs he will need when he uses 3/4 pounds of ground beef would be = 1¹/20 cup.
What are breadcrumbs?Breadcrumbs is a type of food product that is produced from crumbling of dried bread which is used making dishes such as meatballs.
The number of cups of breadcrumbs for 1¼ of meat ball = ¾ cup
Therefore the number of cups of breadcrumbs for 1¾ = X cup.
That is ; 1¼ = ¾ cup
1¾ = X cup
Make X cup the subject of formula;
X cup = 1 ¾ × ¾ ÷ 1¼
X cup = 21/16 ÷ 5/4
X cup = 21/16 × 4/5
X cup = 21/20
X cup = 1 ¹/20 cup
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How many solutions does the following system of equation have y = 2 x + 22y = 4x + 4
Answers
How many solutions does the following system of equation have
y = 2 x + 2
2y = 4x + 4
The answer is : Infinitely many solutions ( because they lie on each other ).