Question

(in standard form)
write a polynomial function of least degree with integral coefficients that has the given zeros.

1. -5, -1, 3
2. 3 + 2i

1. state the zeros of the polynomial. the multiplicity of each zero. the degree of the polynomial and sketch a reasonable graph of the polynomial.

$f(x) = 0.03(x - 5)^2(x + 3)^2$

1. write the factored form of a polynomial equation of least degree to describe the graph.

Explanation:

Problem 16:

Step1: Recall Factor Theorem

If \( r \) is a zero of a polynomial, then \( (x - r) \) is a factor. Given zeros \(-5\), \(-1\), \(3\), the factors are \( (x + 5) \), \( (x + 1) \), \( (x - 3) \).

Step2: Multiply the factors

$$
LATEXBLOCK0
$$

Step1: Complex Conjugate Root Theorem

If a polynomial with real coefficients has a complex zero \( a + bi \), then its conjugate \( a - bi \) is also a zero. Given zero \( 3 + 2i \), the other zero is \( 3 - 2i \).

Step2: Form the polynomial

The factors are \( (x - (3 + 2i)) \) and \( (x - (3 - 2i)) \). Multiply them:

$$
LATEXBLOCK0
$$

Step1: Identify Zeros from Factored Form

For \( f(x) = 0.03(x - 5)^2(x + 3)^2 \), set each factor to zero: \( x - 5 = 0 \) gives \( x = 5 \); \( x + 3 = 0 \) gives \( x = -3 \).

Step2: Determine Multiplicity

The exponent of \( (x - 5) \) is \( 2 \), so multiplicity of \( 5 \) is \( 2 \). The exponent of \( (x + 3) \) is \( 2 \), so multiplicity of \( -3 \) is \( 2 \).

Step3: Find Degree

Sum of exponents: \( 2 + 2 = 4 \), so degree is \( 4 \).

Step4: Sketching the Graph

• End Behavior: Leading term is \( 0.03x^4 \), so as \( x \to \pm\infty \), \( f(x) \to \infty \).
• At \( x = 5 \) (even multiplicity), graph touches x - axis and turns around. At \( x = -3 \) (even multiplicity), graph touches x - axis and turns around.

Answer:

\( f(x)=x^3 + 3x^2 - 13x - 15 \)

Problem 17: