Problem 3 - Algebra 2

Question

Find all of the rational zeros of each function. \(p(x)=x^{3}-5 x^{2}-22 x+56\)

Step-by-Step Solution

Answer

The rational zeros are \(x = 2, 7,-4\).

Step 1: Identify Potential Rational Zeros

According to the Rational Root Theorem, any rational solution of the polynomial equation \(p(x)=x^3-5x^2-22x+56\) is of the form \(\frac{p}{q}\). Here \(p\) is a factor of the constant term 56, and \(q\) is a factor of the leading coefficient 1. So, the potential rational zeros are the factors of 56, which are \(\pm 1, \pm 2, \pm 4, \pm 7, \pm 8, \pm 14, \pm 28, \pm 56\).

Step 2: Test Potential Zeros

Substitute the potential rational zeros into the polynomial function \(p(x)=x^3-5x^2-22x+56\) to find which ones make the function equal zero. Start with simple numbers like \(x = 1\), \(x = -1\), \(x = 2\), etc. After testing, you find:\(p(1) = 1^3 - 5(1)^2 - 22(1) + 56 = 30\) (not zero),\(p(2) = 2^3 - 5(2)^2 - 22(2) + 56 = 0\) (is zero),\(p(7) = 7^3 - 5(7)^2 - 22(7) + 56 = 0\) (is zero).Therefore, \(x = 2\) and \(x = 7\) are rational zeros.

Step 3: Factor the Polynomial Using Known Zeros

Using the known zeros \(x = 2\) and \(x = 7\), we can factor \(p(x)\) as follows: First, divide the polynomial by \((x-2)\) using synthetic division to find the quotient. Then, divide the resulting quotient by \(x-7\) to find the remaining factor.Upon performing synthetic division:1. Divide \(x^3-5x^2-22x+56\) by \(x-2\) to get \(x^2 - 3x - 28\).2. Divide \(x^2 - 3x - 28\) by \(x-7\) to get \(x + 4\).Thus, \(p(x) = (x-2)(x-7)(x+4)\).

Step 4: Determine All Rational Zeros

From the factored form \(p(x) = (x-2)(x-7)(x+4)\), it's clear that the rational zeros of the polynomial are the roots of each factor, which are \(x = 2\), \(x = 7\), and \(x = -4\).

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving variables and coefficients. They are sums of terms, each consisting of a variable raised to a non-negative integer power. In simple terms, it's a function of the form:

• Constant: a term without a variable, e.g., 56 in our function.
• Linear: a term with the variable raised to the power of 1, e.g., -22x.
• Quadratic: includes terms like -5x².
• Cubic, or higher: like x³ in our given polynomial.

The degree of a polynomial is determined by the highest power of the variable. For instance, since the highest degree of the variable in our example is 3, it is a cubic polynomial. Understanding how polynomial functions are structured helps in identifying potential solutions or zeros of these functions.

Factorization of Polynomials

The factorization of a polynomial involves breaking it down into simpler, non-divisible parts called factors. These factors, when multiplied together, recreate the original polynomial. In our example, after discovering potential zeros, we factor the polynomial:

• Start with identifying zeros using the Rational Root Theorem and test these values.
• Two successful tests revealed viable zeros: 2 and 7. Thus, \((x-2)\) and \((x-7)\) are factors.
• Use synthetic division to test the other factors and reduce complexity.
• Finally, further division yielded the last factor \((x+4)\).

Thus, the polynomial \(p(x)\) factors to \((x-2)(x-7)(x+4)\). This process not only helps in simplifying polynomials but also in finding rational zeros, as these zeros are roots where the polynomial equals zero.

Synthetic Division

Synthetic division is a simplified method for dividing a polynomial by a binomial of the form \((x-c)\). It’s less tedious than long division and is particularly useful for testing potential rational roots. Below is a brief step-by-step of using synthetic division:

• List the coefficients of the polynomial. For example, for \(x^3 - 5x^2 - 22x + 56\), the coefficients are 1, -5, -22, and 56.
• Choose a test zero (such as a rational root) and use it with these coefficients.
• Perform operations column-wise: multiply the test zero by the first coefficient and add it to the next.
• Repeat the process across the entire row of coefficients.
• The final number represents the remainder. A zero remainder confirms that the chosen test is a root.

In our example, we used synthetic division twice to confirm both \(x-2\) and \(x-7\) as factors and finally reduce to \((x+4)\), simplifying the polynomial greatly.