Question

For each of the following polynomial equations, determine if \(x=1\) is a root. (a) \(x^{3}-2 x^{2}-3 x-2=0\) (c) \(3 x^{4}-x^{2}+2 x-4=0\) (b) \(2 x^{3}-\frac{1}{2} x^{2}+x-2=0\)

Short answer

(a) No; (b) No; (c) Yes.

Step-by-step solution

Evaluate the First Polynomial at x=1

Substitute \(x = 1\) into the polynomial \(x^3 - 2x^2 - 3x - 2\). Calculate:\[1^3 - 2(1)^2 - 3(1) - 2 = 1 - 2 - 3 - 2\]Simplify:\[1 - 2 - 3 - 2 = -6\]Since \( -6 eq 0\), \(x=1\) is not a root of the polynomial \(x^3 - 2x^2 - 3x - 2\).

Evaluate the Second Polynomial at x=1

Substitute \(x = 1\) into the polynomial \(3x^4 - x^2 + 2x - 4\). Calculate:\[3(1)^4 - (1)^2 + 2(1) - 4 = 3 - 1 + 2 - 4\]Simplify:\[3 - 1 + 2 - 4 = 0\]Since the result is \(0\), \(x=1\) is a root of the polynomial \(3x^4 - x^2 + 2x - 4\).

Evaluate the Third Polynomial at x=1

Substitute \(x = 1\) into the polynomial \(2x^3 - \frac{1}{2}x^2 + x - 2\). Calculate:\[2(1)^3 - \frac{1}{2}(1)^2 + 1 - 2 = 2 - \frac{1}{2} + 1 - 2\]Simplify:\[2 - 0.5 + 1 - 2 = 0.5\]Since \(0.5 eq 0\), \(x=1\) is not a root of the polynomial \(2x^3 - \frac{1}{2}x^2 + x - 2\).

Key concepts

Roots of Polynomials

When tackling polynomial equations, identifying the roots is one of the key tasks. A root, also known as a zero, is a value for which the polynomial equals zero. For instance, if substituting a number into the polynomial results in zero, then that number is a root. In the exercise, we checked whether \(x=1\) was a root for different polynomials. Applying this process generally involves:

• Substituting the suspected root value into the polynomial.
• Simplifying the expression.
• Checking if the resulting value is zero.

If the final result of your operation is zero, the suspected value is indeed a root. Otherwise, it is not. This basic principle helps determine solutions for polynomial equations. The importance here is that roots are fundamental to solving polynomial equations, as they indicate where the function crosses the x-axis.

Polynomial Evaluation

Polynomial evaluation is the process of determining the output of a polynomial function given a specific input. This is done by substituting the input into the polynomial and then simplifying the expression. This process was clearly shown in the step-by-step solution with the evaluation at \(x=1\). Here's how you do it:

• Replace every instance of the variable (usually \(x\)) with the given value.
• Follow the order of operations: parentheses, exponents, multiplication and division, and addition and subtraction.
• Simplify the expression until you reach a result.

Evaluating a polynomial helps determine the value of the polynomial at specific points. It's particularly useful in verifying potential roots or understanding the general behavior of the polynomial across the number line.

Algebraic Concepts

Working with polynomials requires a grasp of fundamental algebraic concepts. These include understanding how to work with exponents, coefficients, and the structure of terms. In algebra, each term of a polynomial is a product of a constant (the coefficient) and a variable raised to a non-negative integer power (the exponent). Here are some vital concepts:

• Variables and Constants: Variables represent unknowns, whereas constants are fixed values that multiply the variables.
• Exponents: They show how many times a variable is multiplied by itself. For example, in \(x^3\), 3 is the exponent indicating \(x\times x\times x\).
• The Polynomial Degree: This is the highest power of the variable in the polynomial, which influences the curve's shape and number of roots.

Understanding these concepts is crucial for solving and simplifying polynomials, identifying possible roots, and understanding their graphical representation on the coordinate plane.