Problem 17 - Precalculus : Building Concepts and Connections

Question

Show that the given value of \(x\) is a zero of the polynomial. Use the zero to completely factor the polynomial. $$p(x)=3 x^{3}+x^{2}+24 x+8 ; x=-\frac{1}{3}$$

Step-by-Step Solution

Answer

The polynomial \(p(x)\) can factor into \((3x+1)(x^{2}+8)\) after demonstrating that \(x=-\frac{1}{3}\) is a zero of the polynomial \(p(x)=3 x^{3}+x^{2}+24 x+8\).

Step 1: Verify Zero

Firstly, check whether \(x=-\frac{1}{3}\) is a zero of the polynomial \(p(x)\) by substituting \(x=-\frac{1}{3}\) into the polynomial. If the result is zero, then it confirms that \(x=-\frac{1}{3}\) is indeed a root of the polynomial.

Step 2: Perform Synthetis Division

Next, use synthetic division to divide the coefficients \(3\), \(1\), \(24\), and \(8\) by \(x=-\frac{1}{3}\). The result will yield the coefficients of the remaining polynomial when \(p(x)\) is divided by \(x+1/3\). This process shows how to factor out \(x+1/3\) from \(p(x)\).

Step 3: Factoring the Polynomial

Finally, write the polynomial factor, which has been achieved by synthetic division. The result will be the factorization of the original polynomial \(p(x)\). If additional zeros are identified, continue the synthetic division process until the polynomial is fully factored.

Key Concepts

Synthetic Division

Understanding the concept of synthetic division can greatly simplify the process of factoring polynomials. Synthetic division is a shorthand method of dividing a polynomial by a binomial of the form (x - c) and is especially useful when the binomial is a linear factor of the polynomial. It involves using only the coefficients of the polynomial, and it's a much quicker and less cumbersome process than long division.

To perform synthetic division, you start by writing down the coefficients of the polynomial you are dividing. Next, you write the zero of the binomial – in our case, the value (-1/3) – outside the bracket that holds the coefficients. You then bring down the first coefficient and multiply it by (-1/3), placing the result under the second coefficient. Continue this process until you reach the end, and the final line of numbers will give the coefficients of the quotient polynomial.

The process can be visualized as follows:

• Write coefficients: 3, 1, 24, 8
• Write the value that 'x' is set equal to on the outside: -1/3
• Perform the synthetic division operation

If there's a remainder after this division, then (-1/3) is not a zero of the polynomial, but if the remainder is 0, we successfully divided the polynomial by (x + 1/3).

Polynomial Zeros

A zero of a polynomial is any value of (x) that, when substituted into the polynomial, will yield a result of zero. In other words, if we have a polynomial (p(x)) and an (x) such that (p(x) = 0), then (x) is considered to be a zero or a root of the polynomial. Identifying the zeros of a polynomial is a crucial step in the factorization process because each zero corresponds to a factor of the polynomial.

For instance, in our original problem with the polynomial (p(x) = 3x^3 + x^2 + 24x + 8), the value (-1/3) is given as a potential zero. To confirm this, we need to substitute (x) with (-1/3) in the polynomial:

• (p(-1/3) = 3(-1/3)^3 + (-1/3)^2 + 24(-1/3) + 8)
• Calculate the value
• Check if the result is 0

Discovering that the value is indeed zero confirms that it is a root, which we can now use for further factoring of the polynomial.

Polynomial Factorization

Polynomial factorization is the process of breaking down a polynomial into the product of simpler polynomials, preferably linear factors when possible. The factored form makes it easier to solve equations and understand the behavior of the graph of the polynomial. Each factor corresponds to a zero of the polynomial, as per the Fundamental Theorem of Algebra, which states that every non-zero single-variable polynomial with complex coefficients has as many complex roots as its degree, counting multiplicity.

Once a zero is identified, such as (-1/3) in our example, we can use this zero to construct a factor of the polynomial. In this case, since (-1/3) is a zero, (x + 1/3) becomes a factor. Using synthetic division or another method to divide the polynomial by this factor gives us a smaller degree polynomial, which we can then analyze further for additional zeros and factors.

The process of repeated factoring and finding zeros continues until the polynomial is completely broken down into linear factors or irreducible quadratic factors. It's important to remember that if a polynomial has real coefficients, any complex zeros will come in conjugate pairs. So, it's often a matter of finding and factoring out one root to uncover additional real or complex roots to simplify the original polynomial entirely.