Problem 18
Question
Show that the given value of \(x\) is a zero of the polynomial. Use the zero to completely factor the polynomial. $$p(x)=2 x^{5}+x^{4}-2 x-1 ; x=-\frac{1}{2}$$
Step-by-Step Solution
Verified Answer
The given value \(x = -1/2\) is a zero of the polynomial \(p(x) = 2x^{5}+x^{4}-2x-1\). This zero can be used to factorize the polynomial as \(p(x) = (x+1/2)(2x^{4}-x^{3}+x^{2}-x+2)\).
1Step 1: Verifying the Value of x
Verify by substituting the value \(x = -1/2\) into the given polynomial \(p(x) = 2x^{5}+x^{4}-2x-1\). If \(p(-1/2) = 0\), then \(x = -1/2\) is a zero of the polynomial.
2Step 2: Performing Synthetic Division
Use the zero to perform synthetic division. Write the coefficients of the polynomial (2, 1, 0, 0, -2, -1) and the zero (-1/2) into a synthetic division table. Carry out the synthetic division process.
3Step 3: Obtaining the Resulting Polynomial
The quotient from the synthetic division is the coefficients of the polynomial that remain after factoring out the term corresponding to the zero. With 5 as degree of our polynomial, means after synthetic division we ended with a quartic polynomial.
4Step 4: Factoring the Polynomial Completely
Rewrite the given polynomial \(p(x) = 2x^{5}+x^{4}-2x-1\) as a product, \(p(x) =(x+1/2)(2x^{4}-x^{3}+x^{2}-x+2)\). The polynomial is now factored completely.
Key Concepts
Synthetic DivisionZero of the PolynomialPolynomial Roots
Synthetic Division
Synthetic Division is a simplified method for dividing a polynomial by a binomial of the form \(x - c\). It makes the division process quicker and easier, eliminating the need for long division. This method uses only the coefficients of the polynomial rather than the entire expression. Let's explore how it works.
First, write down the coefficients of the polynomial in a horizontal line. For the polynomial \(p(x) = 2x^5 + x^4 - 2x - 1\), the coefficients are: 2, 1, 0, 0, -2, and -1. Notice the zeros, which account for missing degrees of \(x^3\) and \(x^2\). Next, write the zero of the polynomial, \(-\frac{1}{2}\), to the left side.
The synthetic division process involves multiplying the current value in a running total by the zero and adding this to the next coefficient. Begin with the first coefficient as the initial value. Multiply by the zero and add to the next coefficient, and so on.
The last value obtained is the remainder. If it's zero, the zero used is indeed a zero of the polynomial, and the other values are the coefficients of the reduced polynomial. This method efficiently helps verify zeros and simplifies polynomials.
First, write down the coefficients of the polynomial in a horizontal line. For the polynomial \(p(x) = 2x^5 + x^4 - 2x - 1\), the coefficients are: 2, 1, 0, 0, -2, and -1. Notice the zeros, which account for missing degrees of \(x^3\) and \(x^2\). Next, write the zero of the polynomial, \(-\frac{1}{2}\), to the left side.
The synthetic division process involves multiplying the current value in a running total by the zero and adding this to the next coefficient. Begin with the first coefficient as the initial value. Multiply by the zero and add to the next coefficient, and so on.
The last value obtained is the remainder. If it's zero, the zero used is indeed a zero of the polynomial, and the other values are the coefficients of the reduced polynomial. This method efficiently helps verify zeros and simplifies polynomials.
Zero of the Polynomial
A Zero of a Polynomial is a value of \(x\) that makes the polynomial equal to zero. In other words, it is the \(x\)-value where the graph of the polynomial intersects the x-axis. Determining the zeros of a polynomial is crucial in algebra because they help in factoring polynomials, understanding graphs, and solving equations.
To confirm if a given number is a zero, substitute it into the polynomial for \(x\). If the polynomial evaluates to zero, then the number is indeed a zero. For example, in the polynomial \(p(x) = 2x^5 + x^4 - 2x - 1\), substituting \(x = -\frac{1}{2}\) results in:
Finding a zero aids in dividing the polynomial and simplifying it to reduce the degree, making it easier to analyze and solve.
To confirm if a given number is a zero, substitute it into the polynomial for \(x\). If the polynomial evaluates to zero, then the number is indeed a zero. For example, in the polynomial \(p(x) = 2x^5 + x^4 - 2x - 1\), substituting \(x = -\frac{1}{2}\) results in:
- First, calculate \(2\left(-\frac{1}{2}\right)^5 + \left(-\frac{1}{2}\right)^4 - 2\left(-\frac{1}{2}\right) - 1\)
- Replacing all \(x\) gives: \(-\frac{1}{16} + \frac{1}{16} + 1 - 1 = 0\)
- Since the result is zero, \(-\frac{1}{2}\) is confirmed as a zero of the polynomial.
Finding a zero aids in dividing the polynomial and simplifying it to reduce the degree, making it easier to analyze and solve.
Polynomial Roots
Polynomial Roots are essentially the solutions of the polynomial equation when it is set to zero. Identifying these roots, or zeros, is a fundamental goal when working with polynomials.
Once a polynomial is reduced using synthetic division, the resulting polynomial can be further factored or solved to find more roots. For instance, after using \(-\frac{1}{2}\) to factor \(p(x) = 2x^5 + x^4 - 2x - 1\), the remaining polynomial is \(2x^4 - x^3 + x^2 - x + 2\). This process can continue through various methods like synthetic division, factoring, or using the quadratic formula for simpler quadratic polynomials.
Each root represents a point where the polynomial crosses or touches the x-axis on a graph. Understanding these points can help visualize the behavior of the polynomial. Knowing all roots and their multiplicities can also reveal information about the polynomial's symmetry and turning points.
Finding all roots of a polynomial is key in many applications across mathematics, helping solve complex equations and interpret mathematical models in practical contexts.
Once a polynomial is reduced using synthetic division, the resulting polynomial can be further factored or solved to find more roots. For instance, after using \(-\frac{1}{2}\) to factor \(p(x) = 2x^5 + x^4 - 2x - 1\), the remaining polynomial is \(2x^4 - x^3 + x^2 - x + 2\). This process can continue through various methods like synthetic division, factoring, or using the quadratic formula for simpler quadratic polynomials.
Each root represents a point where the polynomial crosses or touches the x-axis on a graph. Understanding these points can help visualize the behavior of the polynomial. Knowing all roots and their multiplicities can also reveal information about the polynomial's symmetry and turning points.
Finding all roots of a polynomial is key in many applications across mathematics, helping solve complex equations and interpret mathematical models in practical contexts.
Other exercises in this chapter
Problem 17
Determine what type of symmetry, if any, the function illustrates. Classify the function as odd, even, or neither. $$f(x)=-3 x^{3}+1$$
View solution Problem 18
Solve the polynomial inequality. $$(x+3)\left(x^{2}-3 x+2\right) \geq 0$$
View solution Problem 18
Find all the zeros, real and nonreal, of the polynomial. Then express \(p(x)\) as a product of linear factors. $$p(x)=x^{2}-5$$
View solution Problem 18
Find the domain and the vertical and horizontal asymptotes (if any). $$f(x)=\frac{3 x+5}{x^{2}-x-2}$$
View solution