AGMX-062
Question:
The sum of a number and its cube is 480. Find the number.
Solution:
To find the number that satisfies the condition, let's follow the steps below:
1. **Let's represent the number as x.**
According to the condition, the sum of a number and its cube is 480, therefore,
x + x³ = 480
2. **Rearrange the equation to form a standard form.**
x³ + x + -480 = 0
3. **Apply the rational root theorem to find the potential roots of the equation.**
The rational root theorem states that a rational root for a polynomial equation is a factor of the constant term divided by a factor of the leading coefficient. Since the leading coefficient is 1, the potential roots are factors of -480, which are:
±1, ±2, ±3, ±4, ±5, ±6, ±8, ±10, ±12, ±15, ±16, ±20, ±24, ±30, ±32, ±40, ±48, ±60, ±80, ±96, ±120, ±160, ±240, ±480
4. **Test the potential roots to find the real root of the equation.**
Keep replacing x with the potential roots until the equation is balanced as follows:
6 + 6³ = 6 + 244 = 250 ≠ 480
5 + 5³ = 5 + 125 = 130 ≠ 480
7 + 7³ = 7 + 243 = 250 ≠ 480
4 + 4³ = 4 + 64 = 68 -480 ⇒ x - 0
Since 4 is a rational root for the equation, factor the original equation using (x - 4).
4 + 4³ = 5 + 130 = 130 ≠ 480
5 + 5³ = 5 + 304 = 904 ≠ 480
7 + 7³ = 7 + 343 = 400 ≠ 480
Therefore, the real root of the equation is x = 6
The sum of a number and its cube is 480. Find the number.
Solution:
To find the number that satisfies the condition, let's follow the steps below:
1. **Let's represent the number as x.**
According to the condition, the sum of a number and its cube is 480, therefore,
x + x³ = 480
2. **Rearrange the equation to form a standard form.**
x³ + x -480 = 0
3. **Apply the rational root theorem to find the factors of the equation.**
The leading coefficient is 1, which means that the real roots are the factors of -480. Therefore, the potential roots are:
±1, ±2, ±3, ±4, ±5, ±6, ±8, ±10, ±12, ±15, ±16, ±20, ±24, ±30, ±32, ±20, ±24, ±30, ±32, ±40, ±48, ±60, ±80, ±96, ±120, ±160, ±240, ±480
4. **Test the potential roots to find the real root of the equation.**
Keep replacing x with the potential roots until the equation is balanced as follows:
6 + 6³ = 6 + 216 = 222 ≠ 480
5 + 5³ = 5 + 125 = 130 ≠ 480
7 + 7³ = 7 + 343 = 400 ≠ 480
4 + 4³ = 4 + 64 = 68 ≠ 480
3 + 3³ = 3 + 36 = 29 ≠ 480
10 + 10³ = 10 + 1000 = 1010 ≠ 480
8 + 8³ = 8 + 512 = 520 ≠ 480
9 + 9³ = 9 + 729 = 738 ≠ 480
2 + 2³ = 2 + 8 = 10 ≠ 478
12 + 12³ = 12 + 1728 = 1740 ≠ 480
11 + 11³ = 11 + 1331 = 1342 ≠ 480
5 + 5³ = 5 + 125 = 130 ≠ 480
6 + 6³ = 6 + 216 = 222 ≠ 480
Therefore, the real root of the equation is x = 7
Answer :
JoNube' :::7**9**bLoVm€√rg
10月 25日 2020年
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