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Chapter 1: Problem 97

Find all real solutions \(x\) of the equation $$ x^{10}-x^8+8 x^6-24 x^4+32 x^2-48=0 . $$

Short Answer

Expert verified
The real solutions are \( x = \pm \sqrt{2} \).

Step by step solution

01

Factor out powers of x

Observe that each term in the equation has an even power of x. Let's substitute \( y = x^2 \) to simplify the expression. This gives us \( y^5 - y^4 + 8y^3 - 24y^2 + 32y - 48 = 0 \).
02

Solve for y

Next, solve the polynomial equation \( y^5 - y^4 + 8y^3 - 24y^2 + 32y - 48 = 0 \). Identify any factor patterns or use polynomial solving techniques. After solving, we find that \( y = 2 \) is a repeated root.
03

Substitute back to x

Since \( y = x^2 \), substitute back to get \( x^2 = 2 \).
04

Solve for x

Solve for \( x \) by taking the square root of both sides. This gives \( x^2 = 2 \), so \( x = \pm \sqrt{2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Factorization
Polynomial factorization is a method of breaking down a complicated polynomial into simpler factors that, when multiplied together, give the original polynomial. This process is helpful for solving polynomial equations, as it can reveal the roots of the equation. In our exercise, the polynomial equation was factored by substituting another variable, making it easier to handle. Hence, if you have a polynomial, look for ways to rewrite it as a product of smaller, simpler polynomials. This might involve recognizing certain patterns, like the difference of squares or the sum of cubes, and applying those to factorize effectively.
Substitution Method
The substitution method involves replacing a part of the equation with a new variable to simplify the solving process. In our specific exercise, we used the substitution method by letting \( y = x^2 \). This simplified our original polynomial equation from a degree of 10 to a degree of 5, making it more straightforward to solve. To use substitution effectively:
  • Identify a part of the polynomial that can be substituted for simplification.
  • Replace it with a new variable.
  • Solve the simplified equation.
  • Substitute back the original variable to solve for the actual terms.
This step can transform a seemingly complicated problem into a more manageable one.
Roots of Equations
Finding the roots of an equation helps determine where the polynomial equals zero. For the polynomial \( y^5 - y^4 + 8y^3 - 24y^2 + 32y - 48 = 0 \), the root \( y = 2 \) was found. The roots are the solutions to the polynomial equation and provide valuable information about the behavior of the polynomial. To find the roots:
  • Look for patterns or factorize the polynomial.
  • Use algebraic techniques like synthetic division or the Rational Root Theorem.
  • Verify the roots by substituting them back into the equation.
Understanding the roots gives insight into the nature of the polynomial and the curve it represents.
Real Solutions
A real solution is any solution that is a real number. In our specific problem, the real solutions were \( x = \pm \sqrt{2} \). These solutions are derived from the real roots of the polynomial after substituting back to the original variable. To find real solutions:
  • Ensure the equation is simplified and factored.
  • Solve for the variable using algebraic techniques or substitution.
  • Check if the solutions are real by ensuring they can be expressed as real numbers.
Real solutions are crucial as they represent actual, measurable values and can be plotted or applied directly in real-world scenarios.

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Most popular questions from this chapter

Can there be a multiplicative \(n \times n\) magic square \((n>1)\) with entries \(1,2, \ldots, n^2\) ? That is, does there exist an integer \(n>1\) for which the numbers \(1,2, \ldots, n^2\) can be placed in a square so that the product of all the numbers in any row or column is always the same?

Note that a triangle \(A B C\) is isosceles, with equal angles at \(A\) and \(B\), if and only if the median from \(C\) and the angle bisector at \(C\) are the same. This suggests a measure of "scalenity": For each vertex of a triangle \(A B C\), measure the distance along the opposite side from the midpoint to the "end" of the angle bisector, as a fraction of the total length of that opposite side. This yields a number between 0 and \(1 / 2\); take the least of the three numbers found in this way. The triangle is isosceles if and only if this least number is zero. What are the possible values of this least number if \(A B C\) can be any triangle in the plane?

Find the smallest possible \(n\) for which there exist integers \(x_1, x_2, \ldots, x_n\) such that each integer between 1000 and 2000 (inclusive) can be written as the sum, without repetition, of one or more of the integers \(x_1, x_2, \ldots, x_n\). (It is not required that all such sums lie between 1000 and 2000, just that any integer between 1000 and 2000 be such a sum.)

Let \(a\) and \(d\) be relatively prime positive integers, and consider the sequence \(a, a+d, a+4 d, a+9 d, \ldots, a+n^2 d, \ldots\) Given a positive integer \(b\), can one always find an integer in the sequence which is relatively prime to \(b\) ?

A gambling game is played as follows: \(D\) dollar bills are distributed in some manner among \(N\) indistinguishable envelopes, which are then mixed up in a large bag. The player buys random envelopes, one at a time, for one dollar and examines their contents as they are purchased. If the player can buy as many or as few envelopes as desired, and, furthermore, knows the initial distribution of the money, then for what distribution(s) will the player's expected net return be maximized?
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