Problem 20 - Basic Technical Mathematics with Calculus

Question

Use the factor theorem to determine whether or not the second expression is a factor of the first expression. Do not use synthetic division. $$x^{5}-2 x^{4}+3 x^{3}-6 x^{2}-4 x+8, x-1$$

Step-by-Step Solution

Answer

Yes, \(x-1\) is a factor of the polynomial.

Step 1: Recall Factor Theorem

The factor theorem states that a polynomial has a factor \((x - a)\) if and only if the polynomial evaluates to zero when \(x = a\). In other words, for a polynomial \(f(x)\), \((x - a)\) is a factor if \(f(a) = 0\).

Step 2: Determine Value of 'a'

The second expression \(x-1\) suggests that we should check whether \(x=1\) is a root of the polynomial \(x^5 - 2x^4 + 3x^3 - 6x^2 - 4x + 8\).

Step 3: Substitute x = 1 into the Polynomial

Substitute \(x = 1\) into the polynomial: \[ f(1) = (1)^5 - 2(1)^4 + 3(1)^3 - 6(1)^2 - 4(1) + 8 \].

Step 4: Calculate the Result

Simplify the equation: \[ f(1) = 1 - 2 + 3 - 6 - 4 + 8 \].

Step 5: Simplify Further

Perform the arithmetic step-by-step: - \(1 - 2 = -1\)- \(-1 + 3 = 2\)- \(2 - 6 = -4\)- \(-4 - 4 = -8\)- \(-8 + 8 = 0\).

Step 6: Conclusion from the Factor Theorem

Since \(f(1) = 0\), the factor \((x - 1)\) is indeed a factor of the polynomial \(x^5 - 2x^4 + 3x^3 - 6x^2 - 4x + 8\).

Key Concepts

Polynomials

Polynomials are algebraic expressions composed of variables and coefficients, involving operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. They are fundamental components of algebra and can appear in various forms, such as linear, quadratic, cubic, and higher degree polynomials.
Examples of polynomials include:

• Linear: \(ax + b\)
• Quadratic: \(ax^2 + bx + c\)
• Cubic: \(ax^3 + bx^2 + cx + d\)

In general, a polynomial is expressed as \( f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \), where \( a_n \) through \( a_0 \) are constants, and \( n \) is a non-negative integer representing the degree of the polynomial. The degree indicates the highest power of the variable within the polynomial. Polynomials are widely used not only in mathematics but also in fields like physics, engineering, and economics, where they model various phenomena.

Roots of Polynomials

The roots of a polynomial are the values of the variable that make the polynomial equal to zero. In other words, if \(f(x)\) is a polynomial, then the roots are the solutions to the equation \(f(x) = 0\).
Roots are also known as 'zeros' or 'solutions' and are crucial in understanding the behavior of polynomials. Each root corresponds to a point where the polynomial graph intersects the x-axis.
For example, if you have a polynomial \(f(x) = x^2 - 4\), the roots can be found by solving \(x^2 - 4 = 0\):

• Add 4 to both sides: \(x^2 = 4\).
• Take the square root of both sides: \(x = \pm 2\).

Therefore, \(x = 2\) and \(x = -2\) are the roots. Knowing the roots aids in plotting the graph of the polynomial and in determining certain characteristics like turning points and intercepts.

Algebraic Expressions

Algebraic expressions include constants, variables, and the operations of addition, subtraction, multiplication, and division, excluding division by zero. They form the backbone of algebra and can vary in complexity from simple expressions to intricate polynomial expressions.
Simple algebraic expressions might look like \(2x + 3\) or \(7y - 5\), whereas more complex ones can include polynomials, rational expressions, and radicals. The study of algebraic expressions involves combining like terms, factoring, expanding products, and simplifying for better understanding and use.
For example, to simplify the expression \(2(x + 3) + 4x\), you would:

• First distribute: \(2x + 6 + 4x\)
• Then combine like terms: \(6x + 6\).

Algebraic expressions are solved by setting them equal to a value or another expression and solving for the variable involved. Understanding these concepts is crucial for solving equations and inequalities and for further exploration in advanced mathematics like calculus.