Problem 48 - A Graphical Approach to College Algebra

Question

Use synthetic division to determine whether the given number is a zero of the polynomial. $$-4 ; \quad P(x)=9 x^{3}+39 x^{2}+12 x$$

Step-by-Step Solution

Answer

-4 is a zero of the polynomial.

Step 1: Set Up Synthetic Division

To use synthetic division, list the coefficients of the polynomial: 9, 39, 12, and 0 (for the missing constant term). These go in a row. The number to the left, outside the synthetic division symbol, will be -4, which we are testing as a zero.

Step 2: Perform Synthetic Division

Start by bringing down the leading coefficient (9) directly. Multiply -4 by this coefficient (9), getting -36, and place it under the next coefficient (39). Add these two numbers to get 3, which becomes the new number in the row. Repeat this process: multiply -4 by 3 to get -12, add this to the next coefficient (12) to get 0. Finally, multiply -4 by 0 to get 0, and add to the constant term (0) to get 0.

Step 3: Analyze the Remainder

The last number in the synthetic division row is the remainder. If this number is 0, then -4 is a zero of the polynomial. In this case, the remainder is 0.

Step 4: Conclude the Result

Since the remainder is 0, -4 is a zero of the polynomial, meaning that P(-4) = 0.

Key Concepts

Polynomials

Polynomials are a key part of algebra and mathematics in general. They are expressions that consist of variables and coefficients. These expressions can have one or more terms:

• The variables are usually represented by letters like \(x\).
• The coefficients are numbers that multiply the variables.
• A polynomial is usually written in the form of \(a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\).
• Each term in the polynomial consists of a coefficient, the variable, and an exponent.

In the given exercise, the polynomial is \(P(x)=9x^3+39x^2+12x\). Here, 9, 39, and 12 are the coefficients and the constant term is missing.

Zeros of Polynomials

Zeros of polynomials are values of the variable that make the polynomial equal to zero. In simple terms, if you plug in that value for the variable, the entire expression should result in zero:

• To find the zeros, set the polynomial equal to zero and solve for the variable.
• Zeros are also known as roots of the polynomial.
• Understanding the zeros is crucial because they give insights into the polynomial's behavior, such as where the graph crosses the x-axis.

In the step-by-step solution, synthetic division was used to test if \(-4\) is a zero of the polynomial \(P(x)\). Since the remainder was zero, \(-4\) is indeed a zero of the polynomial.

Division of Polynomials

Dividing polynomials can seem challenging, but techniques like synthetic division simplify the process considerably.

• Synthetic division is a shortcut method specifically used when dividing by linear polynomials of the form \(x-c\).
• It involves less writing and fewer calculations, which makes it quicker than long division for many problems.
• In this exercise, dividing the polynomial \(9x^3+39x^2+12x\) by \(x+4\) reduced to using \(-4\) synthetically because of the opposite sign rule.

The steps involve using the coefficients, performing multiplications, and then adding to determine the new coefficients and remainder.

Remainder Theorem

The remainder theorem is a useful concept in polynomial division. It states that when a polynomial \(P(x)\) is divided by \(x-c\), the remainder of this division is \(P(c)\). This theorem helps us find the remainder without performing the entire division:

• If the remainder is 0, then \(x-c\) is a factor of the polynomial, and \(c\) is a zero.
• In the context of our problem, the remainder was indeed zero.
• This confirmed that \(-4\) is a zero of the polynomial \(P(x)\).

Using this theorem can save a lot of time and is especially useful for testing potential zeros quickly.