Problem 69 - A Graphical Approach to College Algebra

Question

Use the rational zeros theorem to completely factor \(P(x)\). $$P(x)=21 x^{4}+13 x^{3}-103 x^{2}-65 x-10$$

Step-by-Step Solution

Answer

The polynomial is fully factored as \((x+1)(3x-1)(7x^2+x+10)\).

Step 1: Understanding the Problem

We need to factor the polynomial \(P(x) = 21x^4 + 13x^3 - 103x^2 - 65x - 10\) completely by using the Rational Zeros Theorem.

Step 2: Identify Possible Rational Zeros

According to the Rational Zeros Theorem, any rational root \( \frac{p}{q} \) of \(P(x)\) is such that \(p\) divides the trailing constant term (-10) and \(q\) divides the leading coefficient (21). The possible rational zeros are fractions of the factors of -10 over 21.

Step 3: List Factors of Constant and Leading Coefficients

The factors of -10 are \(\pm 1, \pm 2, \pm 5, \pm 10\) and the factors of 21 are \(\pm 1, \pm 3, \pm 7, \pm 21\).

Step 4: Form Possible Rational Zeros

Combine the factors to form possible rational zeros: \(\pm 1, \pm 2, \pm 5, \pm 10, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{5}{3}, \pm \frac{10}{3}, \pm \frac{1}{7}, \pm \frac{2}{7}, \pm \frac{5}{7}, \pm \frac{10}{7}, \pm \frac{1}{21}, \pm \frac{2}{21}, \pm \frac{5}{21}, \pm \frac{10}{21}\).

Step 5: Evaluate Potential Zeros

Utilize synthetic division or direct substitution to evaluate these candidates and find the actual roots of the polynomial.

Step 6: Find a Zero

After testing several values, we find that \(x = -1\) is a zero of the polynomial.

Step 7: Use Synthetic Division

Perform synthetic division using \(x+1\) to divide the polynomial \(P(x)\) by \(x+1\). The result should be a cubic polynomial \(Q(x)\).

Step 8: Further Factor the Result

Continue factoring \(Q(x)\) using methods such as factoring by grouping, synthetic division, or the use of additional theorems.

Step 9: Complete Factorization

Repeat the process until \(Q(x)\) is broken down into all of its linear factors.

Step 10: Express the Factorized Form

The completely factored form of \(P(x)\) is \((x+1)(3x-1)(7x^2+x+10)\).

Key Concepts

Polynomial Factoring

Polynomials are mathematical expressions consisting of variables and coefficients. Factoring a polynomial involves breaking it down into simpler terms (or factors) that, when multiplied together, yield the original polynomial. This process helps simplify complex equations, making solutions more accessible.

Factors of a polynomial are expressions of lesser degree. For example, to factor a quartic polynomial like our example, we begin by finding an expression of a lower degree that divides the initial polynomial evenly. Typically, this means identifying binomials or other polynomials that can multiply to produce the original polynomial.

• Factoring helps solve polynomial equations by simplifying calculations.
• It's done by expressing the polynomial as a product of its factors.
• The degree of the factors should sum up to the degree of the original polynomial.

Mastering this fundamental algebraic skill makes it easier to solve polynomial equations and find their zeros or roots.

Synthetic Division

Synthetic division is a quick method of dividing a polynomial by a binomial of the form \(x - c\). Easier than long division, it's especially handy when determining polynomial factors or evaluating potential zeros. This tool is invaluable for working with polynomials because it allows us to efficiently perform division operations.

In our example, once we discover a candidate zero, such as \(x = -1\), we use synthetic division to simplify the division under this assumption.

The steps are:

• Write down the coefficients of the polynomial.
• Use the potential zero in the division process, altering these coefficients through addition and multiplication following specific rules.
• Confirm if the remainder is zero; this indicates that the divisor was indeed a factor.

This method's speed and simplicity make it an excellent choice for factorization and polynomial division.

Rational Roots

The Rational Zeros Theorem is a fundamental principle that guides us in finding possible rational solutions to polynomial equations. According to this theorem, any rational solution \(\frac{p}{q}\) of a polynomial is such that \(p\) divides the constant term, and \(q\) divides the leading coefficient.

By comprehending this theorem, we can create a list of potential rational roots or zeros to test within the polynomial. This drastically narrows down the possibilities, making it easier to manage the testing process.

In our case, the divisors of the constant term (-10) and the leading coefficient (21) provide our necessary factors to predict possible rational zeros:

• The factors of -10 are \(\pm 1, \pm 2, \pm 5, \pm 10\).
• The factors of 21 are \(\pm 1, \pm 3, \pm 7, \pm 21\).

Combining these in various ways gives us potential solutions. Testing these possibilities sequentially leads us towards identifying true zeros.

Algebra Problem Solving

Tackling algebra problems, especially with polynomials, requires a systematic approach. Each problem-solving scenario, like finding zeros or factoring polynomials, involves several steps and methods. Understanding these methods aids in solving complex algebra problems more effectively.

Here are some strategies to consider:

• Break down the problem: Start by understanding what needs to be done, such as finding factors or roots.
• Use specific theorems like Rational Zeros and synthetic division: These tools are tailored for polynomials and can simplify the process.
• Test and verify: Once potential solutions are found, verify by plugging the roots back into the polynomial.
• Stay organized: Keep track of factors and divide systematically to avoid mistakes.

By mastering these approaches, solving algebra problems involving polynomials becomes a more manageable task, fostering deeper comprehension and better academic performance.