Prove that: (a2 - b2)3 + (b2-c2)3+ (c2-a2)3 = 3 (a+b) (b+c) (c+a) (a-b) (b-c) (c-a).

Answers

Answer 1

Answer: $L=(a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3=(*)\n\n(a^2-b^2)^3=a^6-3a^4b^2+3a^2b^4-b^6\n\n(b^2-c^2)^3=b^6-3b^4c^2+3b^3c^4-c^6\n\n(c^2-a^2)^3=c^6-3c^4a^2+3c^2a^4-a^6\n\n(*)=a^6-3a^4b^2+3a^2b^4-b^6+b^6-3b^4c^2+3b^2c^4-c^6+c^6-\dots\n\dots-3c^4a^2+3c^2a^4-a^6\n\n=-3a^4b^2+3a^2b^4-3b^4c^2+3b^2c^4-3c^4a^2+3c^2a^4\n\n=3(-a^4b^2+a^2b^4-b^4c^2+b^2c^4-a^2c^4+a^4c^2)$

$R=3(a+b)(a-b)(b+c)(b-c)(c+a)(c-a)\n\n=3(a^2-b^2)(b^2-c^2)(c^2-a^2)\n\n=3(a^2b^2-a^2c^2-b^4+b^2c^2)(c^2-a^2)\n\n=3(a^2b^2c^2-a^4b^2-a^2c^4+a^4c^2-b^4c^2+a^2b^4+b^2c^4-a^2b^2c^2)\n\n=3(-a^4b^2+a^2b^4-b^4c^2+b^2c^4-a^2c^4+a^4c^2)\n\nL=R$

Answer 2

Answer:

$(a^2 - b^2)^3 + (b^2 - c^2)3 + (c^2 - a^2)^3 = 3(a + b)(b +c)(c + a)(a - b)(b - c)(c - a)$
$(a^2 - b^2)^3 + (b^2 - c^2)3 + (c^2 - a^2)^3 = 3(a + b)(a - b)(b + c)(b - c)(c + a)(c - a)$
$(a^2 - b^2)^3 + (b^2 - c^2)3 + (c^2 - a^2)^3 = 3(a^2 - b^2)(b^2 - c^2)(c^2 - a^2)$

$(a^2 - b^2)^3 = (a^2 - b2)(a^2 - b^2)(a^2-b^2) = a^6 - 3a^4b^2 + 3a^2b^4 - b^6$
$(b^2 - c^2)^3 = (b^2 - c^2)(b^2 - c^2)(b^2 - c^2) = b^6 - 3^4c^2 + 3b^2c^4 - c^6)$
$(c^2 - a^2)^3 = (c^2 - a^2)(c^2 - a^2)(c^2 - a^2) = c^6 - 3a^2c^4 + 3a^4c^2 - a^6$

$a^6 - 3a^4b^2 + 3a^2b^4 - b^6 + b^6 - 3b^4c^2 + 3b^2c^4 - c^6 + c^6 - 3a^2c^4 + 3a^4c^2 - a^6$
$-3a^4b^2 + 3a^2b^4 - 3b^4c^2 + 3b^2c^4 - 3a^2c^4 + 3a^4c^2$
$3(-a^4b^2 + a^2b^4 - b^4c^2 + b2^c^4 - a^2c^4 + a^4c^2)$

$3(a^2 - b^2)(b^2 - c^2)(c^2 - a^2) = 3(-a^4b^2 + a^2b^4 - b^4c^2 + b^2c^4 - a^2c^4 + a^4c^2)$

$3(-a^4b^2 + a^2b^4 - b^4c^2 + b^2c^4 - a^2c^4 + a^4c^2) = 3(-a^4b^2 + a^2b^4 - b^4c^2 + b^2c^4 - a^2c^4 + a^4c^2$

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What are the zeros of the polynomial function: f(x) = x3 + x2 – 6x ? ...?

Answers

The solution to the problem is as follows:

　
x³ + x² − 6x = 0
x (x² + x − 6) = 0
x (x − 2) (x + 3) = 0
x = 0, 2, −3

f(x) = 2x³ + 5x² + 6x − 4
f(−x) = −2x³ + 5x² − 6x − 4

Let's look at signs only and how many sign changes there are:
f(x) = + + + −
1 sign change ----> 1 positive root
f(−x) = − + − −
2 sign changes ----> 2 or 0 negative roots

Min. number of real roots = 1 ----> max. number of complex roots = 2
Max. number of real roots = 3 ----> min. number of complex roots = 0

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What is the value of the expression 9×4−(2×12)? Enter your answer in the box.

Answers

Answer:

Step-by-step explanation:

9x4-(2x12)

= 36-24

= 12

If f(x) = 2x + 3 and g(x) = x2 + 1,find f(g(3)).

Answers

Answer:

The value of f(g(3)) is 23.

Step-by-step explanation:

The given functions are

$f(x)=2x+3$

$g(x)=x^2+1$

We have to find the value of f(g(3)).

Substitute x=3 in the function g(x) to find the value of g(3).

$g(3)=3^2+1=9+1=10$

$f(g(3))=f(10)$ [g(3)=10]

Substitute x=10 in the function f(x) to find the value of f(10).

$f(10)=2(10)+3=20+3=23$

Therefore the value of f(g(3)) is 23.

Hello,

f(g(3))=f(3²+1)=f(10)=2*10+3=23
===========================

A straight line has gradient m and passes through the points (0,-2). Find the two values for m for which the line is tangent to the curve y=x^2-2x 7. And for each value of m, find the coordinates of the points where the line touches the curve

Answers

Answer:

when m=2 or -2, the line is tangent to the curve at the points (2.7) and (0,7)

Step-by-step explanation:

Answer:

Hi,

Step-by-step explanation:

The curve is y=x^²-2x+7

The tangent has for equation: y=mx-2

Let's find the intersection of the tangent and the curve.

$mx-2=x^2-2x+7\n\nx^2+(-2-m)x+9=0\n\Delta=(-2-m)^2-4*9=m^2+4m-32\nThe\ point\ must\ be \ unique: \Delta=0\n\nm^2+4m-32=0\n\delta=4^2+4*32=144=12^2\n\nm=(-4-12)/(2) =-8\ or\ m=(-4+12)/(2) =4\n\nx^2+(-2-m)x+9=0\ and\ \Delta=0 = > x=(2+m)/(2) \n\nif\ m=-8\ then\ x=(2+(-8))/(2)=-3\ and\ y=mx-2=(-8)*(-3)-2=22\n\nif\ m=4\ then\ x=(2+4)/(2)=3\ and\ y=mx-2=4*3-2=10\n\n\nCoordinates\ of\ the\ points\ are (-3,22)\ and\ (3,10)\n$

Out of 1,100 discs tested, 13 are defective. Estimate the number of defective discs in a batch of 41,000.

Answers

This is a simple proportion

1100 : 13 = 41000 : x

For every 1100 discs, 13 are defective, meaning that there will be an X number for each 41000

13 * 41000 = 1100 * x

13 * 410 = 11 * x
5330 = 11 * x
x = 5330/11
x = 484.5

That means that for every 41000, approximately 485 discs will break down.

Answer:

x = 484.5

That means that for every 41000, approximately 485 discs will break down

The table below shows the relationship between the number of miles traveled, x, and the number of gallons of gas used, y.х
35
70
105
140
175
Y
1
2
3
4
5
Which of the following equations best represents the relationship?
A. y = 3.5x
B. y = 35x
C. y = 1/35x
D. 35 = 1x

Answers

The relationship between the number of miles traveled, x, and the number of gallons of gas used, y is, y = 35x . So Option B is correct

What is the equation of line?

The equation of a straight line is a relationship between x and y coordinates, The general equation of a straight line is y = mx + c, where m is the slope of the line and c is the y-intercept.

Given that,

the number of miles traveled x, the number of gallons of gas used, y

35, 1

70, 2

105, 3

140, 4

175, 5

For 1 mile distance, 35 gallons of gas is required,

And for 2 mile distance, 70 gallons gas is required

So, the number of gallons y = 35 times the number of miles traveled

Now it can be write as,

Y = 35X

Hence, the equation is Y = 35X

To know more about Line check:

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Answer: (y = 35x)

Step-by-step explanation: you get your answer to y which is the number of gallons used and x is the number of miles traveled; Meaning 35x (35 miles traveled) is y (1 gallon used).

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