What is the degree of the polynomial?

Determine all prime numbers a, b and c for which the expression a ^ 2 + b ^ 2 + c ^ 2 - 1 is a perfect square .

kogti [31]

Answer:

The family of all prime numbers such that $a^{2} + b^{2} + c^{2} -1$ is a perfect square is represented by the following solution:

$a$ is an arbitrary prime number. (1)

$b = \sqrt{1 + 2\cdot a \cdot c}$ (2)

$c$ is another arbitrary prime number. (3)

Step-by-step explanation:

From Algebra we know that a second order polynomial is a perfect square if and only if $(x+y)^{2} = x^{2} + 2\cdot x\cdot y + y^{2}$ . From statement, we must fulfill the following identity:

$a^{2} + b^{2} + c^{2} - 1 = x^{2} + 2\cdot x\cdot y + y^{2}$

By Associative and Commutative properties, we can reorganize the expression as follows:

$a^{2} + (b^{2}-1) + c^{2} = x^{2} + 2\cdot x \cdot y + y^{2}$ (1)

Then, we have the following system of equations:

$x = a$ (2)

$(b^{2}-1) = 2\cdot x\cdot y$ (3)

$y = c$ (4)

By (2) and (4) in (3), we have the following expression:

$(b^{2} - 1) = 2\cdot a \cdot c$

$b^{2} = 1 + 2\cdot a \cdot c$

$b = \sqrt{1 + 2\cdot a\cdot c}$

From Number Theory, we remember that a number is prime if and only if is divisible both by 1 and by itself. Then, $a, b, c > 1$ . If $a$ , $b$ and $c$ are prime numbers, then $2\cdot a\cdot c$ must be an even composite number, which means that $a$ and $c$ can be either both odd numbers or a even number and a odd number. In the family of prime numbers, the only even number is 2.

In addition, $b$ must be a natural number, which means that:

$1 + 2\cdot a\cdot c \ge 4$

$2\cdot a \cdot c \ge 3$

$a\cdot c \ge \frac{3}{2}$

But the lowest possible product made by two prime numbers is $2^{2} = 4$ . Hence, $a\cdot c \ge 4$ .

The family of all prime numbers such that $a^{2} + b^{2} + c^{2} -1$ is a perfect square is represented by the following solution:

$a$ is an arbitrary prime number. (1)

$b = \sqrt{1 + 2\cdot a \cdot c}$ (2)

$c$ is another arbitrary prime number. (3)

Example: $a = 2$ , $c = 2$

$b = \sqrt{1 + 2\cdot (2)\cdot (2)}$

$b = 3$

4 0

3 years ago

WHAT IS THE LINE OF BEST FIT FOR THE FOLLOWING DATA (1,3) (2,0) (5,-8) (9, -25) . WRITE YOUR ANSWERS IN THE FORM OF Y= AX+ B

dimulka [17.4K]

Answer:

3.471 + 7.2516 x

Step-by-step explanation:

Given the data:

(1,3) (2,0) (5,-8) (9, -25) .

X:

1

2

5

9

Y:

3

0

-8

-25

The line of best fit for the data given can be obtained using an online regression calculator :

The equation of best fit for the data is thus;

Best fit equation; y = -3.471 + 7.2516x

Intercept = - 3.471 ;

Slope = 7.2516

5 0

3 years ago

5. A chef prepared 48,462 plates of shrimp. There were 96 shrimp on each plate. How many

marta [7]

Answer:

504 I think

Step-by-step explanation:

48,462 divided by 96

KEYWORD: Each, each always means division

4 0

3 years ago

Read 2 more answers

What is the equation of the axis of symmetry for the parabola y= 1/4(x+2)^2-7

Nezavi [6.7K]

Your answer is -2 as the axis of symmetry is at -b/2a. So if you were to expand the equation, you would get y= x^2/4 +x -6 where b is 1 and 2a is ½ . Therefore -1 / ½ gives you -2

7 0

3 years ago

(PLEASE HELP/NO SPAM) Which equation shows the function y=2^x translated 4 units left and 2 units down?

lianna [129]

Answer:

The answer is (h).

Step-by-step explanation:

When y = 2^x is shifted 4 units left, the x shifts 4 units to the left and x becomes (x+4). When the graph is shifted to units down, y has to be subtracted by 2.

The function becomes

y = 2^(x+4) -2.

Finally, you shift -2 to the same side as y.

This becomes

y+2 = 2^(x+4) which is (h)

5 0

3 years ago