Problem 71 - Algebra 2

Question

Construct a polynomial function with the given zeros. $$ x=0,1,4 $$

Step-by-Step Solution

Answer

The polynomial function with the given zeros 0, 1 and 4 is \(x^3 - x^2 - 4x\).

Step 1: Identify the Factors

Given that 0, 1 and 4 are zeros, the corresponding factors of the polynomial function would be \(x - 0\), \(x - 1\), and \(x - 4\) respectively.

Step 2: Form the Polynomial

Multiply the factors to form the polynomial. Hence, the polynomial will be \((x - 0) * (x - 1) * (x - 4)\).

Step 3: Simplify the Polynomial

The polynomial can be simplified by multiplying the factors. Expression \(x - 0\) simplifies to \(x\). The full multiply of the expression gives the polynomial \(x * (x - 1) * (x - 4) = x (x^2-x-4) = x^3 - x^2 - 4x\). Here, \(x^3 - x^2 - 4x\) is the required polynomial.

Key Concepts

Zeros of a Polynomial

Zeros of a polynomial once identified, play a crucial role in constructing the polynomial function. The zero or root of a polynomial is a value for which the polynomial evaluates to zero. For example, if you have a polynomial function \(f(x)\), and \(f(a)=0\), then \(a\) is a zero of the polynomial.

• Each zero of the polynomial corresponds to a factor of the polynomial (i.e., \((x - a)\) if \(a\) is a zero).
• If you are given zeros like \(0, 1, 4\), then the factors of the polynomial are \((x - 0), (x - 1), (x - 4)\).

Zeros help in understanding the points where the graph of the polynomial will intersect the x-axis. Identifying zeros is the first step in forming a polynomial based on given conditions, and they help in reconstructing the polynomial's expression.

Factoring Polynomials

Factoring polynomials involves breaking down the polynomial into simpler expressions called factors that, when multiplied together, give back the original polynomial. This process is essential in simplifying polynomial equations and solving them efficiently.

• Each factor corresponds to a solution or zero of the polynomial equation.
• For instance, based on the zeros \(0, 1, 4\), the factored form would be \((x - 0)(x - 1)(x - 4)\).

Knowing how to factor polynomials effectively aids in both understanding the polynomial's structure and in graphing it. It reduces complex problems into more manageable parts, shedding light on possible simplifications. Sometimes, special techniques such as grouping or using the quadratic formula are employed to assist in the factoring process of more complex polynomials.

Simplifying Polynomials

Simplifying a polynomial involves reducing it to its simplest form. This step is usually about expanding the factors through multiplication to form a polynomial that contains no brackets or parentheses.

• Starting with factors like \((x - 0)(x - 1)(x - 4)\), you can simplify the expression by first addressing \(x - 0\), which simplifies directly to \(x\).
• Next, continue by expanding: \(x * (x - 1)\) and then take the result and multiply by \((x - 4)\).

The simplified expression would be \(x(x^2 - x - 4) = x^3 - x^2 - 4x\).
Simplifying provides an elegant version of the polynomial. It enables easier graphing and analysis while revealing the polynomial's degree and leading coefficient. Through simplification, you'll get a clearer view of the function's overall behavior and graphical representation.