Problem 40 - Precalculus with Calculus Previews

Question

Find the zeros of the given polynomial function \(f .\) State the multiplicity of each zero. $$ f(x)=x^{4}+6 x^{3}+9 x^{2} $$

Step-by-Step Solution

Answer

The zeros are 0 (multiplicity 2) and -3 (multiplicity 2).

Step 1: Factor out the greatest common factor

The given polynomial function is \( f(x) = x^4 + 6x^3 + 9x^2 \). Notice that each term in the polynomial contains \(x^2\) as a factor. So, factor out \(x^2\):\[ f(x) = x^2(x^2 + 6x + 9) \].

Step 2: Factor the quadratic expression

Look at the quadratic expression \( x^2 + 6x + 9 \). This is a perfect square trinomial that can be factored as:\[ x^2 + 6x + 9 = (x + 3)^2 \]. Thus, the polynomial becomes:\[ f(x) = x^2 (x + 3)^2 \].

Step 3: Identify the zeros and their multiplicities

To find the zeros, set \( f(x) = 0 \):\[ x^2 (x + 3)^2 = 0 \]. Solving \( x^2 = 0 \) gives \( x = 0 \) with a multiplicity of 2. Solving \( (x + 3)^2 = 0 \) gives \( x = -3 \) with a multiplicity of 2.

Key Concepts

Multiplicity of Zeros

In understanding the concept of zeros of a polynomial, it's important to also grasp the concept of **multiplicity**. Multiplicity refers to the number of times a particular zero appears in a polynomial. This is determined by the exponent of the factor associated with the zero.

Consider a polynomial equation factored as

• In this example, the polynomial is given as: \( f(x) = x^2(x + 3)^2 \).
• The zero \( x = 0 \) comes from the factor \( x^2 \) and has a multiplicity of 2 since the factor is squared.
• Similarly, the zero \( x = -3 \) arises from the factor \( (x + 3)^2 \) and also has a multiplicity of 2.

The multiplcity of zeros can indicate the nature of the graph at these points. For instance:

• If the multiplicity is odd, the graph will cross the x-axis at the zero.
• If the multiplicity is even, the graph touches the x-axis and turns around at the zero.

Understanding multiplicity helps predict the shape and behavior of the graph of the polynomial.

Factoring Polynomials

Factoring polynomials is a crucial skill in algebra. It involves expressing a polynomial as a product of its factors, which simplifies solving equations to find the zeros of the polynomial function.

In our given polynomial \( f(x) = x^4 + 6x^3 + 9x^2 \), we start by looking for the **greatest common factor (GCF)** among all terms:

• Identify the common factor in each term, which in this case is \( x^2 \).
• Factor out \( x^2 \), simplifying the polynomial to \( x^2(x^2 + 6x + 9) \).

Next, we need to factor the remaining quadratic expression, which can further simplify the polynomial:

• The quadratic \( x^2 + 6x + 9 \) is recognized as a perfect square trinomial.
• It is factored into \( (x + 3)^2 \), making the final factored form \( f(x) = x^2(x + 3)^2 \).

Factoring not only assists in finding zeros but also aids in simplifying calculations and solving polynomial equations.

Perfect Square Trinomials

Perfect square trinomials are special quadratic expressions that are formed by squaring a binomial. Recognizing these can simplify the factoring process greatly.

A perfect square trinomial takes the form \( a^2 + 2ab + b^2 \) and can be rewritten as \((a + b)^2\). In our case:

• We identified \( x^2 + 6x + 9 \) as a perfect square trinomial.
• It is rewritten as \((x + 3)^2\), where \( x \) is \( a \), and \( 3 \) is \( b \).

Recognizing perfect square trinomials can greatly ease the process of factoring because:

• It converts a three-term expression into a simpler binomial squared form.
• This makes identifying zeros and their multiplicity straightforward.

Getting comfortable with these patterns can save time and reduce errors when working with polynomial equations.