Problem 59 - Algebra and Trigonometry

Question

Find a polynomial of the specified degree that has the given zeros. Degree \(4 ; \quad\) zeros \(-1,1,3,5\)

Step-by-Step Solution

Answer

The polynomial is \( f(x) = x^4 - 8x^3 + 14x^2 + 8x - 15 \).

Step 1: Use the Zeros Formula for a Polynomial

To find a polynomial with specified zeros, we can use the fact that if its degree is equal to the number of zeros and all the zeros are real, then the polynomial can be expressed as a product of linear factors of the form \( f(x) = a(x - r_1)(x - r_2)(x - r_3)...(x - r_n) \), where \( a \) is a non-zero constant and the \( r_i \) are the roots (zeros).

Step 2: Write the Linear Factors

With the given zeros \(-1, 1, 3, 5\), the linear factors of the polynomial are \((x + 1), (x - 1), (x - 3), (x - 5)\). Thus, the polynomial can be expressed as: \( f(x) = a(x + 1)(x - 1)(x - 3)(x - 5) \).

Step 3: Expand the Linear Factors

Expand \((x + 1)(x - 1)\) and \((x - 3)(x - 5)\) separately:- \( (x + 1)(x - 1) = x^2 - 1 \)- \( (x - 3)(x - 5) = x^2 - 8x + 15 \)

Step 4: Multiply the Results of the Expansions

Now multiply the two quadratic expressions obtained:\[ (x^2 - 1)(x^2 - 8x + 15) \]Use the distributive property to expand:\[ x^4 - 8x^3 + 15x^2 - x^2 + 8x - 15 \]

Step 5: Simplify the Polynomial

Combine like terms in the expanded expression:\[ x^4 - 8x^3 + 14x^2 + 8x - 15 \] Since the problem doesn't specify any particular leading coefficient, we can assume \( a = 1 \). Thus, the polynomial is \( f(x) = x^4 - 8x^3 + 14x^2 + 8x - 15 \).

Key Concepts

Zeros of Polynomials

When dealing with polynomials, understanding the concept of "zeros" is crucial. The zeros of a polynomial, also known as the "roots," are the values of the variable that make the polynomial equal to zero. In other words, if you substitute a zero back into the polynomial equation, the result will be zero.

For example, if a polynomial is given by \( f(x) = (x + 1)(x - 1)(x - 3)(x - 5) \), the zeros are the solutions \( x = -1, 1, 3, \) and \( 5 \). These are the values of \( x \) that satisfy \( f(x) = 0 \).

To find these zeros from a polynomial that's already expanded, factor the polynomial back into a product of linear factors, as the factors directly give the zeros. This is an important step in solving equations and analyzing graphs of polynomial functions.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial expression. It is an important concept because it provides insight into the behavior and characteristics of the polynomial function.

For instance, the polynomial constructed as \( f(x) = (x + 1)(x - 1)(x - 3)(x - 5) \) is of degree 4. This is evident because when multiplying the four linear factors, the highest power of \( x \) in the expression will be \( x^4 \).

The degree also tells us about the maximum number of real roots (or zeros) the polynomial can have, which is equal to the degree itself, assuming there are no complex or repeated zeros. Additionally, the degree provides information about the general shape of the graph, such as how many times it can curve.

Linear Factors

Linear factors are expressions of the form \( (x - r) \), where \( r \) is a root (zero) of the polynomial. Composing a polynomial through its linear factors is a fundamental technique in algebra that helps in finding and verifying the zeros quickly.

If you know the zeros of a polynomial, you can write the polynomial as a product of these linear factors. For a polynomial with zeros at \(-1, 1, 3, \) and \(5\), the linear factors are \((x + 1), (x - 1), (x - 3), \) and \((x - 5)\). Each factor corresponds to moving the term from zero on the number line.

This representation is useful for expanding, simplifying, and understanding the intrinsic properties of the polynomial function, such as its degree, zeros, and even its graph.

Expanding Polynomials

Expanding a polynomial means writing it as a sum of terms rather than a product of factors. This is done through multiplication and the distributive property. It may seem tedious, but it allows you to work with the polynomial in a more straightforward way for simplifications and other calculations.

For instance, expanding \( (x + 1)(x - 1) \) results in \( x^2 - 1 \), and \( (x - 3)(x - 5) \) gives \( x^2 - 8x + 15 \).

Next, we multiply these two quadratic polynomials: \((x^2 - 1)(x^2 - 8x + 15)\). Using the distributive property, multiply each term in the first polynomial by each term in the second polynomial.

• This results in: \( x^4 - 8x^3 + 15x^2 - x^2 + 8x - 15 \).

After that, combine like terms to simplify the expression: \( x^4 - 8x^3 + 14x^2 + 8x - 15 \). This is the completely expanded form of the original polynomial, revealing both its structure and behavior clearly.