Problem 64 - Algebra and Trigonometry

Question

Find a polynomial of degree 4 that has integer coefficients and zeros \(1,-1,2,\) and \(\frac{1}{2} .\)

Step-by-Step Solution

Answer

The polynomial is \(2x^4 - 5x^3 + 5x - 2\).

Step 1: Identify the Roots

The given zeros of the polynomial are 1, -1, 2, and \(\frac{1}{2}\). We need to construct a polynomial with these roots. Since we want a polynomial with integer coefficients and one of the roots is a fraction, we will need to consider the conjugate or related factor to keep coefficients integers.

Step 2: Form Factors from Roots

The roots give us factors. For the integer roots, the factors are \((x - 1)\), \((x + 1)\), and \((x - 2)\). For the root \(\frac{1}{2}\), multiply by 2 to form the integer-coefficient factor \((2x - 1)\).

Step 3: Form the Polynomial

Multiply all the factors together to form the polynomial. This gives us:\[f(x) = (x - 1)(x + 1)(x - 2)(2x - 1)\]

Step 4: Expand the Polynomial

First, multiply the binomials:1. \((x - 1)(x + 1) = x^2 - 1^2 = x^2 - 1\)2. \((x - 2)(2x - 1) = (2x^2 - 4x - x + 2) = 2x^2 - 5x + 2\)Now multiply these results:\((x^2 - 1)(2x^2 - 5x + 2)\)

Step 5: Perform Polynomial Multiplication

Expand \((x^2 - 1)(2x^2 - 5x + 2)\):\[= x^2(2x^2 - 5x + 2) - 1(2x^2 - 5x + 2) = 2x^4 - 5x^3 + 2x^2 - 2x^2 + 5x - 2 = 2x^4 - 5x^3 + 5x - 2 \]

Step 6: Write the Final Polynomial

The polynomial of degree 4 with integer coefficients and the given zeros is:\[f(x) = 2x^4 - 5x^3 + 5x - 2\]

Key Concepts

Integer coefficients

When discussing polynomials, integer coefficients mean that every number in front of a variable (such as the 2 in \(2x^4\)) is a whole number. This is crucial because it ensures a tighter, more predictable set of solutions and behaviors in the polynomial. For our polynomial of degree 4, one of the roots is \(\frac{1}{2}\). However, having a fraction as a root can lead to coefficients that are not integers when derived directly from factors like \((x - \frac{1}{2})\). To resolve this, algebraic manipulation is used to transform such fractional factors into integer-coefficient factors. By multiplying the factor \((x - \frac{1}{2})\) by 2, we obtain \((2x - 1)\), ensuring all the coefficients remain integers. This is a key step, turning potential fractions into manageable whole numbers.

Polynomial roots

Roots of a polynomial are the values of \(x\) that make the entire polynomial equal to zero. For a polynomial of degree 4, you should have typically four roots, accounting for possible multiplicities. The roots can be real or complex and represent the points where the graph of the polynomial will intersect the x-axis.

• In this exercise, the given roots were: \(1\), \(-1\), \(2\), and \(\frac{1}{2}\).
• Each of these has corresponding factors, \((x - 1)\), \((x + 1)\), \((x - 2)\), and \((2x - 1)\).

By identifying these roots, one can not only construct polynomial expressions but also better visualize and understand the function's behavior. The way roots are managed affects subsequent polynomial calculations and manipulations.

Factorization

Factorization is the process of breaking down a complex expression into simpler, more manageable parts (factors) that, when multiplied together, give the original polynomial. For a polynomial, this involves expressing it as a product of its factors based on its roots.This exercise uses the roots \(1\), \(-1\), \(2\), and \(\frac{1}{2}\) to form factors \((x - 1)\), \((x + 1)\), \((x - 2)\), and \((2x - 1)\). Factorization follows directly from these zeroes:

• Each root leads directly to one factor of the polynomial.
• The integer root factors are simple to form: \((x - \text{root})\).
• The fraction is dealt with by multiplying to get a factor that stays within integer coefficients \((2x - 1)\).

Mastering factorization is immensely helpful for simplifying expressions and solving polynomial equations. It allows students to recognize patterns in algebra and perform polynomial division.

Polynomial expansion

Expanding a polynomial means multiplying together its factors to express it in standard polynomial form, which involves several steps of distribution. The initial factors, such as \((x - 1)(x + 1)\) and \((x - 2)(2x - 1)\), need to be multiplied through equity and distribution.For example:

• First simplify \((x - 1)(x + 1)\): It follows the difference of squares formula resulting in \(x^2 - 1^2 = x^2 - 1\).
• For \((x - 2)(2x - 1)\), you expand by distributing each term one by one: \(2x^2 - 5x + 2\).

Then, these conclusions are multiplied: \[(x^2 - 1)(2x^2 - 5x + 2)\]Conducting this leads you to an expanded polynomial form, finishing with steps involving distribution visible in each of the operations chained together. The goal of expansion is to clarify the full expression, showcasing it as a homogeneous, streamlined polynomial ready for graphing or further algebraic manipulation.