Problem 10 - Algebra 2

Question

Write a polynomial function of least degree with integral ceefficients the zeros of which include \(\frac{1}{2}, 3,\) and \(-3 .\)

Step-by-Step Solution

Answer

The polynomial is \(2x^3 - x^2 - 18x + 9\).

Step 1: Identify Roots and Write Factors

The roots given are \(\frac{1}{2}\), \(3\), and \(-3\). To write them as factors, note that each root corresponds to a factor of the polynomial: \((x - \frac{1}{2})\), \((x - 3)\), and \((x + 3)\).

Step 2: Adjust Factor for Non-integral Root

Since polynomial coefficients must be integers and we have a non-integral root \(\frac{1}{2}\), multiply the factor \((x - \frac{1}{2})\) by 2 to eliminate the fraction. This gives \((2x - 1)\) as the factor corresponding to \(\frac{1}{2}\).

Step 3: Compose Polynomial with Integral Coefficients

Now, the polynomial is \((2x - 1)(x - 3)(x + 3)\). Multiply these factors together to form the polynomial of least degree with integral coefficients.

Step 4: Multiply Factors

First, multiply \((x - 3)\) and \((x + 3)\) which yields \((x^2 - 9)\) because it takes the form of a difference of squares: \(a^2 - b^2 = (a-b)(a+b)\).

Step 5: Final Polynomial Expression

Next, multiply \((2x - 1)\) by the result \((x^2 - 9)\). When you do this multiplication step-by-step: \(2x(x^2) + 2x(-9) - 1(x^2) - 1(-9)\) results in \(2x^3 - 18x - x^2 + 9\). Rearrange to make it easier to read as \(2x^3 - x^2 - 18x + 9\).

Key Concepts

Zeroes of Polynomials

In polynomial functions, a zero (or root) is a value of the variable that makes the polynomial equal to zero. When solving problems involving polynomial zeros, it's crucial to remember this connection: each zero corresponds to a factor of the polynomial. For instance, if 3 is a zero, the corresponding factor will be

• (x - 3).

This is because substituting 3 into

• x - 3 gives zero.

If a polynomial has zeros

• a, b,

and

• c,

the factors will be

• (x - a), (x - b), and (x - c).

Understanding zeros helps in composing the polynomial from its roots quickly. Knowing how to convert zeros into polynomial factors can simplify creating and solving polynomials. It's important to follow this factorization method correctly, particularly when dealing with non-integral roots.

Least Degree Polynomial

The degree of a polynomial is determined by its highest power of the variable. For example, if the polynomial is expressed as

• ax^n + bx^{n-1} + ...

, the degree is

• n,

because that's the highest exponent of

• x.

To achieve a polynomial of the least degree with specified zeros, simply multiply the linear factors together without introducing any unnecessary complexity. Each unique zero contributes a linear factor and thus adds one to the polynomial's degree. The exercise modeled here involved the zeros

• \(\frac{1}{2}, 3, -3\),

which leads us to include exactly three linear factors:

• (2x - 1), (x - 3), and (x + 3).

Multiplying these factors gives a polynomial of minimum degree 3, as each factor is of degree 1, and they share no repeated factors.

Integral Coefficients

When creating a polynomial, especially in algebra-based tasks, it's often critical for the polynomial to have integer coefficients. This requirement makes it possible to use these polynomials in various applications without encountering fractions, which could complicate computation.In this exercise, one root was a fraction, \(\frac{1}{2}\). To handle this, we modified the factor

• (x - \frac{1}{2})

to

• (2x - 1).

By multiplying the factor by 2, we eliminated fractions, thus ensuring integer coefficients. If we do not resolve non-integer coefficients, the polynomial might not perform as expected or fit the format required for the problem's solution.Always verify each step of the multiplication to guarantee the final polynomial

• 2x^3 - x^2 - 18x + 9

meets the condition of having integral coefficients consistently throughout. This meticulous attention to detail ensures the polynomial not only solves the mathematical problem but also aligns with real-world computation needs.