Question

Find a polynomial function whose real zeros and degree are given. Answers will vary depending on the choice of the leading coefficient. $$ \text { Zeros: }-1, \text { multiplicity } 1 ; 3, \text { multiplicity } 2 ; \text { degree } 3 $$

Short answer

P(x) = k(x + 1)(x - 3)^2

Step-by-step solution

Identify the Roots

The roots (zeros) given are -1 with multiplicity 1 and 3 with multiplicity 2.

Write the Factors

Since the root -1 has a multiplicity of 1, it corresponds to the factor (x + 1). The root 3 with a multiplicity of 2 corresponds to the factor (x - 3)^2.

Construct the Polynomial

Since the polynomial has a degree of 3, the polynomial function P(x) can be written as: P(x) = k(x + 1)(x - 3)^2, where k is the leading coefficient.

Expand the Polynomial

Expand the expression: (x - 3)^2 = (x - 3)(x - 3) = x^2 - 6x + 9. Therefore,P(x) = k(x + 1)(x^2 - 6x + 9).

Simplify the Expression

Distribute (x + 1) in P(x):P(x) = k(x^3 - 5x^2 + 3x - 9). This is the polynomial function in expanded form.

Key concepts

Real Zeros

In polynomial functions, real zeros are the values of x that make the function equal to zero. They are also called the roots or solutions of the polynomial equation. To find real zeros, set the polynomial equal to 0 and solve for x. For example, if a polynomial function has zeros at -1 and 3, it means substituting -1 or 3 for x in the polynomial equation will result in 0. Knowing the real zeros helps us write the polynomial in factored form, making it easier to study and understand its behavior.

Degree of Polynomial

The degree of a polynomial is the highest power of x in its expression. It tells us the most significant behavior of the polynomial as x becomes very large or very small. In this exercise, the degree is 3.

For example, if we have P(x) = k(x + 1)(x - 3)^2, the highest power of x after expanding will be x³. Therefore, the polynomial is of degree 3.

The degree also indicates the maximum number of real zeros and turning points the polynomial can have. For a 3rd degree polynomial, there can be up to 3 real zeros and up to 2 turning points.

Multiplicity of Roots

Multiplicity refers to the number of times a particular root appears. For example, if the root 3 has a multiplicity of 2, it means that (x - 3) appears twice as a factor.

In our polynomial, -1 has a multiplicity of 1 and 3 has a multiplicity of 2. Hence, our factors are (x + 1) and (x - 3)². Multiplicity affects the graph of the polynomial; roots with odd multiplicity cross the x-axis, while roots with even multiplicity touch it but do not cross.

Leading Coefficient

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. In the polynomial P(x) = k(x + 1)(x - 3)², the leading coefficient is k.

Choosing the value of k affects the stretch or compression of the graph but does not affect the zeros. In this particular problem, we can choose any non-zero value for k. For simplicity, we often set k = 1, so the polynomial remains P(x) = (x + 1)(x - 3)². However, if k ≠1, our polynomial might look like P(x) = 2(x + 1)(x - 3)² or P(x) = −1(x + 1)(x - 3)², each making the graph stretch differently.

Expanding Expressions

To simplify polynomial expressions, we sometimes need to expand them. Expansion involves using the distributive property to multiply factors together. Let's expand (x - 3)²:

(x - 3)(x - 3) = x² - 3x - 3x + 9 = x² - 6x + 9.

Now, P(x) = (x + 1)(x² - 6x + 9). We need to distribute (x + 1) across this expanded form:

(x + 1)(x² - 6x + 9) = x(x² - 6x + 9) + 1(x² - 6x + 9) = x³ - 6x² + 9x + x² - 6x + 9 = x³ - 5x² + 3x - 9.

Thus, P(x) = x³ - 5x² + 3x - 9, the expanded form of the polynomial.