Four consecutive integers have a product of 360 Find the integers by writing a plynomial equation that represents the integers and then solving algebraically.

Question

Four consecutive integers have a product of 360 Find the integers by writing a plynomial equation that represents the integers and then solving algebraically.

Answer 1

let the numbers be

x-1 x x+1 x+2
then
x(x-1)(x+1)(x+2) = 360

since the numbers are all about equal, and since 256 = 4^4, I'd say the numbers are

3,4,5,6

algebraically,
x^4+2x^3-x^2-2x-360 = 0

you know the roots are all factors of 360, and you'd like to try numbers near 4, some synthetic division will quickly show that the solutions are 3,4,5,6.

Answer 2

Sure! Let's start by representing the four consecutive integers as x, x+1, x+2, and x+3. To find their product, we can set up the equation:

x(x+1)(x+2)(x+3) = 360

Expanding this equation, we get:

x(x+1)(x+2)(x+3) - 360 = 0

Now, solving this polynomial equation algebraically, we find that the four consecutive integers are:

x = 5, x+1 = 6, x+2 = 7, and x+3 = 8

So, the four consecutive integers whose product is 360 are 5, 6, 7, and 8.

Answer 3

Let's call the first integer x. Then the second integer would be x + 1, the third integer would be x + 2, and the fourth integer would be x + 3.

The product of these four consecutive integers can be represented by the equation:
x(x + 1)(x + 2)(x + 3) = 360

To solve this equation algebraically, we can simplify it and then solve for x. Let's expand the equation:
(x^2 + 3x)(x^2 + 3x + 2) = 360

Now, let's distribute and simplify:
(x^2 + 3x)(x^2 + 3x + 2) = 360
x^4 + 6x^3 + 11x^2 + 6x = 360

Rearranging the equation to bring all terms to one side:
x^4 + 6x^3 + 11x^2 + 6x - 360 = 0

Now, we need to solve this polynomial equation to find the values of x. We can either factor or use numerical methods to find the values of x. Let's use numerical methods.

Using a calculator or software program, we can find that the solutions of this equation are x = -6, x = -3, x = 4, and x = 5.

So the four consecutive integers are:
-6, -5, -4, and -3
or
-3, -2, -1, and 0
or
4, 5, 6, and 7

Answer 4

To find the four consecutive integers with a product of 360, we can represent the integers algebraically.

Let's assume the first integer is x. Since we want consecutive integers, the second integer will be (x+1), the third integer will be (x+2), and the fourth integer will be (x+3).

To represent the product of these consecutive integers, we can multiply them together:

x * (x+1) * (x+2) * (x+3) = 360

Now we have an equation that represents the problem.

To solve this equation algebraically, we can start by simplifying the equation:

(x^2 + 3x)(x^2 + 3x + 2) = 360

Now, let's multiply it out and set the equation equal to zero:

x^4 + 6x^3 + 11x^2 + 6x - 360 = 0

Unfortunately, this equation is a quartic equation, which can be tricky to solve algebraically. To solve it exactly, it would require factoring or using more advanced methods like the rational root theorem or completing the square.

However, we can use numerical methods or a graphing calculator to find approximate solutions. By graphically plotting the equation and finding the x-intercepts or using a numerical solver, we can determine the values of x.