Problem 67 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find a polynomial function that has the given zeros. (There are many correct answers.) \(0,-2,-4\)

Step-by-Step Solution

Answer

The polynomial function that has the given zeros \(0,-2,-4\) is \(x^3 + 6x^2 + 8x\).

Step 1: Identify the roots

Identify the roots given in the question, these are \(0, -2, -4\) . Each of these roots corresponds to a factor in the polynomial function.

Step 2: Formulate the polynomial

Formulate the polynomial by introducing the factors that correspond to the given roots. Each factor will take the form of \(x - a\), where 'a' is the root. Since the roots are \(0, -2, -4\) , the factors will be \(x, (x+2), (x+4)\). If we multiply these factors together, we get the polynomial \(x(x+2)(x+4)\)

Step 3: Expand the factors

We then need to expand the polynomial. First, multiply the first two factors together: \(x * (x + 2) = x^2 + 2x\). Then, multiply this result by the third factor: \((x^2 + 2x) * (x + 4)\) equals \(x^3 + 6x^2 + 8x\).

Key Concepts

Roots of a Polynomial

Understanding the roots or zeros of a polynomial is essential for grasping how these mathematical objects behave. A root of a polynomial is a number that makes the polynomial equal to zero when it is substituted in place of the variable. In other words, if you plug in the root into the polynomial, the output will be zero. This is a critical aspect because it tells us where the graph of the polynomial intersects the x-axis.

For instance, a polynomial function that includes zeroes at 0, -2, and -4 means the graph touches or crosses the x-axis at these points. Visually, thinking of roots as the places where the curve hits the x-axis can help students better understand the concept. In algebraic terms, each root ‘a’ can be represented by a factor of the form \(x - a\). Therefore, the polynomial \(x(x+2)(x+4)\) must equal zero when \(x\) is 0, -2, or -4.

Factoring Polynomials

Factoring polynomials is a process by which we express a polynomial as the product of its factors - simpler polynomials whose multiplication gives the original polynomial. Factoring is a powerful tool for simplifying expressions and solving equations. To factor a polynomial, you search for expressions that you can multiply together to get back to the original polynomial. These expressions are closely related to the polynomial's roots.

For the case of our example with the roots 0, -2, and -4, we construct factors associated with each root: \(x\) for the root 0, \(x+2\) for the root -2, and \(x+4\) for the root -4. We then combine these factors through multiplication to form the polynomial. Factoring is reversible; that is, you can go from factored form back to expanded form (and vice versa), which leads us right into polynomial expansion.

Polynomial Expansion

Polynomial expansion is the process of multiplying out the factors of a polynomial to return to the standard or expanded form. While a factored form of a polynomial, such as \(x(x+2)(x+4)\), clearly shows the roots, the expanded form can be easier to use for other operations, like differentiation or integration.

It can be a bit tricky to expand polynomials, especially as they get larger. A good method is to do it in steps: first multiply the first two factors, then multiply the result by the next factor, and continue until all factors are multiplied. Using the distributive property, also known as the FOIL (First, Outer, Inner, Last) method for binomials, can simplify the process. For our example, expanding \(x(x+2)(x+4)\) gives us \(x^3 + 6x^2 + 8x\), which is the polynomial in its expanded form. Remember, when expanding, every term in each binomial must multiply by every term in the other binomial(s).