Question

Find a polynomial with minimum degree and leading coefficient 1 that has zeros \(x=3(\) multiplicity 2\(), x=0\) (multiplicity 3 ), and \(x=-2\) (multiplicity 1).

Short answer

The polynomial is \(x^6 - 4x^5 - 3x^4 + 18x^3\).

Step-by-step solution

Identify the Zeros and Their Multiplicities

The given zeros are: - Zero at \(x = 3\) with multiplicity 2 - Zero at \(x = 0\) with multiplicity 3 - Zero at \(x = -2\) with multiplicity 1

Write the Factors Corresponding to Each Zero

Each zero and its multiplicity can be written as a factor in polynomial form: - \( (x - 3)^2 \) for the zero \( x = 3 \) with multiplicity 2 - \( x^3 \) for the zero \( x = 0 \) with multiplicity 3 - \( (x + 2) \) for the zero \( x = -2 \) with multiplicity 1

Form the Polynomial by Multiplying All Factors

Combine all the factors to form the polynomial: \[ (x - 3)^2 \times x^3 \times (x + 2) \]

Expand the Polynomial

Expand the expression to find the polynomial: First, expand \((x - 3)^2\): \[ (x - 3)^2 = x^2 - 6x + 9 \] Next, multiply by \( x^3 \): \[ x^3 \times (x^2 - 6x + 9) = x^5 - 6x^4 + 9x^3 \] Finally, multiply by \( (x + 2) \): \[ (x^5 - 6x^4 + 9x^3) \times (x + 2) \] Expand step by step: \[ x^6 - 6x^5 + 9x^4 + 2x^5 - 12x^4 + 18x^3 \] Combine like terms: \[ x^6 - 4x^5 - 3x^4 + 18x^3 \]

Key concepts

zeros and multiplicities

In polynomial functions, zeros, also called roots, are the values of x where the polynomial equals zero. Each zero can have a multiplicity, which indicates how many times that zero appears as a root. For example, if a polynomial has a zero at \( x = 3 \) with multiplicity 2, it means \( (x - 3) \) is a factor of the polynomial and it appears twice. Similarly, for a zero at \( x = 0 \) with multiplicity 3, \( x \) is a factor occurring three times. Here's a summary of the multiplicities:

• Zero at \( x = 3 \) with multiplicity 2: written as \( (x - 3)^2 \)
• Zero at \( x = 0 \) with multiplicity 3: written as \( x^3 \)
• Zero at \( x = -2 \) with multiplicity 1: written as \( (x + 2) \)

Understanding zeros and their multiplicities is crucial when forming and factoring polynomials.

polynomial expansion

Polynomial expansion involves multiplying the factors of a polynomial to express it as a sum of terms. In our example, we start with the factors: \[ (x - 3)^2 \times x^3 \times (x + 2) \]
We expand each part step by step. First, expand \( (x - 3)^2 \):
\[ (x - 3)^2 = x^2 - 6x + 9 \]
Next, multiply by \( x^3 \):
\[ x^3 \times (x^2 - 6x + 9) = x^5 - 6x^4 + 9x^3 \]
Finally, multiply by \( (x + 2) \):
\[ (x^5 - 6x^4 + 9x^3) \times (x + 2) \]
Expanding this step-by-step:
\[ x^6 - 6x^5 + 9x^4 + 2x^5 - 12x^4 + 18x^3 \]
Combine like terms to get the final polynomial:
\[ x^6 - 4x^5 - 3x^4 + 18x^3 \]
Polynomial expansion is essential for expressing polynomials in their standard form.

factoring

Factoring is the process of breaking down a polynomial into simpler polynomials, called factors, whose product is the original polynomial. Factors are often simpler expressions like \( (x - a) \), where \( a \) is a root of the polynomial. In our exercise, the factors corresponding to the given roots and their multiplicities are:

• \( (x - 3)^2 \) for the zero \( x = 3 \) with multiplicity 2
• \( x^3 \) for the zero \( x = 0 \) with multiplicity 3
• \( (x + 2) \) for the zero \( x = -2 \) with multiplicity 1

Combining these factors:
\[ (x - 3)^2 \times x^3 \times (x + 2) \] Factoring helps in simplifying polynomials and finding their roots more easily.