Questions: Find a polynomial function f(x) with the given degree, zeros, and leading coefficient: Degree 4, x=-3, multiplicity of 2; x=1 multiplicity of 1; x=3, multiplicity of 1 and a leading coefficient of 1. a) The factored form is: b) Find the polynomial equation in standard form. f(x)=

Solution Steps

To find a polynomial function with the given degree, zeros, and leading coefficient, we start by using the zeros and their multiplicities to construct the factored form of the polynomial. Each zero \( x = a \) with multiplicity \( m \) contributes a factor of \( (x - a)^m \) to the polynomial. Once we have the factored form, we expand it to get the polynomial in standard form. The leading coefficient is used to ensure the polynomial has the correct leading term.

Given the zeros and their multiplicities, we can express the polynomial \( f(x) \) in its factored form. The zeros are:

• \( x = -3 \) with multiplicity \( 2 \)
• \( x = 1 \) with multiplicity \( 1 \)
• \( x = 3 \) with multiplicity \( 1 \)

Thus, the factored form of the polynomial is: \[ f(x) = (x + 3)^2 (x - 1)(x - 3) \]

Next, we expand the factored form to obtain the polynomial in standard form. The expansion yields: \[ f(x) = x^4 + 2x^3 - 12x^2 - 18x + 27 \]

Final Answer