Problem 47 - Algebra for College Students

Question

Establish the best integral bounds for the roots of each equation according to the theorem on bounds. $$x^{4}-5 x^{2}+7=0$$

Step-by-Step Solution

Answer

The integral bounds for the roots are -7 to 8.

Step 1: Identify the polynomial

The given polynomial is: \[ f(x) = x^4 - 5x^2 + 7 \]

Step 2: Identify the coefficients

The coefficients of the polynomial \[ f(x) = ax^n + bx^{n-1} + cx^{n-2} + \text{...} \text{ are } 1, 0, -5, 0, 7. \]

Step 3: Theorem on bounds for polynomial roots

The theorem states that if \[ f(x) = a_n x^n + a_{n-1} x^{n-1} + \text{...} + a_0 \] with \[ a_n > 0 \], then all roots lie within \[ -\frac{M}{a_n} \text{ and } 1 + \frac{M}{a_n} \], where M is the maximum absolute value of the coefficients except \[ a_n \].

Step 4: Finding the maximum absolute coefficient

The coefficients are: 1, 0, -5, 0, 7. The maximum absolute value of these coefficients (excluding the leading coefficient, which is 1) is 7.

Step 5: Applying the boundary theorem

Using the boundary theorem: For upper bound: \[ 1 + \frac{7}{1} = 8 \]. And for the lower bound: \[ -\frac{7}{1} = -7 \].

Step 6: Final bounds

Thus, the roots of the polynomial equation \[ x^4 - 5x^2 + 7 = 0 \] lie within the integral bounds: \[ [-7, 8] \].

Key Concepts

theorem on bounds

The theorem on bounds for polynomial roots is a very useful tool in algebra. It helps us determine the range within which all the roots of a polynomial lie. By knowing this range, we can focus our search for roots within a specific interval, making the problem much easier to solve.
The theorem states that if we have a polynomial of the form:
\ \ \f(x) = a_n x^n + a_{n-1} x^{n-1} + \text{...} + a_0, \
with the leading coefficient \( a_n \) being positive, then all roots of this polynomial will lie within: \ \ \ -\frac{M}{a_n} \text{ and } 1 + \frac{M}{a_n}, \
where \( M \) is the maximum absolute value among the polynomial's coefficients, excluding \( a_n \).

For example, in the polynomial \( f(x) = x^4 - 5x^2 + 7 \), the coefficients are 1, 0, -5, 0, and 7. The maximum absolute value among these is 7. Applying the theorem, the roots must lie between -7 and 8.

polynomial roots

Polynomial roots are the values of \( x \) that make the polynomial equal to zero. In other words, if you substitute these values into the polynomial equation, the result will be zero.
Finding the roots of a polynomial can help us understand the behavior of the polynomial, such as where it intersects the x-axis.

• The roots can be real or complex numbers.
• Some polynomials may have multiple roots, also known as multiplicity.
• The degree of the polynomial (highest power of \( x \)) tells us the maximum number of roots.

In our example, the polynomial is \( f(x) = x^4 - 5x^2 + 7 \). We aim to find the values of \( x \) where this polynomial equals zero. Using the bounds theorem, we know that these roots lie between -7 and 8.

algebraic equations

Algebraic equations involve finding the values of variables that satisfy given expressions. The polynomial equation \( x^4 - 5x^2 + 7 = 0 \) is an algebraic equation.
Solving such equations often involves techniques like:

• Factoring the polynomial.
• Using the quadratic formula (for second-degree polynomials).
• Numerical methods for higher-degree polynomials.

In our example, we recognized that traditional methods might be challenging due to the higher degree of the polynomial. Therefore, using the bounds theorem narrows down our search to a specific interval, aiding in either further analytical solutions or numerical methods for finding roots.

integral bounds

Integral bounds refer to the specific range or interval within which some property lies—in this case, the roots of our polynomial. By calculating the integral bounds, we can focus our efforts when solving the polynomial.
For the polynomial \( f(x) = x^4 - 5x^2 + 7 \):

• The coefficients are: 1, 0, -5, 0, 7.
• The maximum absolute value of these coefficients, excluding the leading coefficient, is 7.

Applying the theory on bounds:

• The upper bound is \( 1 + \frac{7}{1} = 8 \).
• The lower bound is \( -\frac{7}{1} = -7 \).

Therefore, all roots of the polynomial equation lie within the integral bounds of [-7, 8]. This means the roots are confined to a specific interval, making them easier to locate.