Problem 23 - Precalculus: Functions and Graphs

Question

Show that the number is a zero of \(f(x)\) of the given multiplicity, and express \(f(x)\) as a product of linear factors. $$\begin{aligned} f(x)=x^{4}+7 x^{3}+13 x^{2}-3 x-18 & \\ &-3 \text { (multiplicity 2) } \end{aligned}$$

Step-by-Step Solution

Answer

\(f(x) = (x + 3)^2 (x^2 + 4x - 2)\) with \(-3\) having multiplicity 2.

Step 1: Plug in the Given Zero

The first step is to verify that \(-3\) is indeed a zero of the function \(f(x)\). To do this, substitute \(x = -3\) into \(f(x)\):\[f(-3) = (-3)^4 + 7(-3)^3 + 13(-3)^2 - 3(-3) - 18\] Calculate the value to ensure \(f(-3) = 0\).

Step 2: Perform Synthetic Division

Since \(x = -3\) is a zero of multiplicity 2, it implies \(x + 3\) is a factor. Perform synthetic division of \(f(x)\) by \(x + 3\) twice to verify the multiplicity and reduce \(f(x)\). First division gives us a quotient polynomial. Repeat the division with the quotient until we reach a quadratic expression.

Step 3: Verify Multiplicity and Factor Further

After performing synthetic division twice, verify the multiplicity by ensuring each division results in zero remainder. The resulting quotient should be a quadratic polynomial if \(-3\) is truly a zero of multiplicity 2.

Step 4: Express as Product of Linear Factors

Factor the resulting quadratic polynomial (from Step 2) into linear factors, if possible. The complete expression for \(f(x)\) as a product of linear factors should include the factor \((x + 3)^2\) to show the zero of multiplicity 2.

Step 5: Write Final Expression

Combining all the factors, write \(f(x)\) as a product of linear factors:\[f(x) = (x + 3)^2 (x^2 + 4x - 2)\]If the quadratic can further be factored into linear factors, do so.

Key Concepts

Synthetic Division

Synthetic division is a streamlined way of dividing polynomials, especially useful when you want to test potential zeros of a function.
Unlike long division, synthetic division is faster and involves fewer steps. It's usually applied when the divisor is a linear factor of the form \( x - c \).

• First, list the coefficients of the polynomial you're dividing.
• The number you're dividing by must be of opposite sign to what appears in the factor.
• The process involves bringing down the first coefficient, multiplying it by the divisor, adding to the next coefficient, and repeating.

For the given polynomial \( f(x) = x^4 + 7x^3 + 13x^2 - 3x - 18 \) and zero \(-3\), perform synthetic division by \(x + 3\).
This will confirm \(-3\) is a zero.
Then, repeat the process to address the multiplicity.

Multiplicity of Zeros

Multiplicity refers to the number of times a particular zero appears in a polynomial. A zero of multiplicity 2, for example, implies the zero appears twice.
Multiplicities affect the function's graph.
For instance, a zero with even multiplicity will touch the x-axis but not cross it. Conversely, an odd multiplicity zero will cross the x-axis.

• To verify a zero's multiplicity, perform synthetic division. Multiple successful divisions with zero remainders signify the multiplicity.
• In this problem, \(-3\) is stated as a zero with multiplicity 2, meaning we can divide \(f(x)\) by \((x + 3)^2\).
• Each division reduces the degree of the polynomial by 1, thus after two divisions, the polynomial degree reduces by 2.

When dividing twice by \((x + 3)\), the remainder being zero both times verifies the zero's multiplicity.

Polynomial Factoring

Factoring polynomials involves expressing them as a product of linear factors, which can often be used to find zeros or roots.
The goal in this exercise is to break down \(f(x)\) completely by factoring out known elements.

• After dividing the polynomial using synthetic division, the remaining polynomial is a quadratic.
• The known zero multiplies into the quadratic, aiding further factoring.

For \(f(x) = (x + 3)^2(x^2 + 4x - 2)\), first establish the multiplicity zero factor as \((x + 3)^2\).
Then, look to factor \(x^2 + 4x - 2\):
Find two numbers that multiply to \(-2\) and add to \(4\).
If no numbers fit, use the quadratic formula to find potential linear factors as roots.
Thus, complete the expression as a product of linear factors.