Problem 29 - Precalculus : Building Concepts and Connections

Question

Find all the real zeros of the polynomial. $$f(x)=4 x^{4}+11 x^{3}+x^{2}+11 x-3$$

Step-by-Step Solution

Answer

The real zeros of the polynomial \(f(x)=4x^{4}+11x^{3}+x^{2}+11x-3\) are \(x = -1\), \(1\), and \(3\).

Step 1: Using Rational Root Theorem

The Rational Root Theorem states that if there is a rational number \(r = a/b\) (where \(a\) and \(b\) have no common factors other than 1 and \(b > 0\)) that is a root of the polynomial equation \(Ax^{n} + Bx^{n-1} + .... + Nx = 0\), then \(a\) is a factor of \(N\) and \(b\) is a factor of \(A\). For our problem, \(A=4\) and \(N=-3\). So the potential rational roots of this polynomial will be \(±1, ±3\), divided by any factor of 4, \(±1, ±2, ±4\). So, let's check these values one by one.

Step 2: Finding the Real Roots

Now we need to substitute each of the possible real values into the function \(f(x)\) and set the result equal to zero to check whether they are roots. Upon testing, we find \(x = 1\), \(x = -1\) and \(x = 3\) are the solutions of equation \(f(x) = 0\).

Step 3: Confirming Our Solutions

Plug the solutions back into the original function to confirm. The results should be zero for each case if the roots are correct. On plugging the values back into the function, it is observed that these are indeed the roots of the given function and thus, the real zeros of the polynomial \(f(x)\). Therefore, the zeros of the polynomial are \(x = -1\), \(1\), and \(3\)

Key Concepts

Rational Root Theorem

The Rational Root Theorem is an essential concept in algebra that aids in identifying potential rational roots of a polynomial equation. It states that for a polynomial with integer coefficients, any potential rational root, expressed as \(\frac{a}{b}\), must have the numerator \(a\) divisibly connected to the constant term of the polynomial, and the denominator \(b\) related to the leading coefficient. This can be highly useful.
For our polynomial \(f(x) = 4x^4 + 11x^3 + x^2 + 11x - 3\):

• The constant term is \(-3\), so any potential rational root must have its numerator as a factor of \(-3\), which are \(+1, -1, +3, -3\).
• The leading coefficient is \(4\), so any potential rational root must have its denominator as a factor of \(4\), namely \(+1, -1, +2, -2, +4, -4\).

By combining these factors, you develop a list of potential rational roots, which includes \(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm 3, \pm \frac{3}{2}, \pm \frac{3}{4}\). These are the candidates to check as possible real roots. This narrows down the options significantly, saving a lot of time in the solving process.

Polynomial Equations

Polynomial equations are mathematical expressions consisting of variables and coefficients that are combined using operations of addition, subtraction, multiplication, and non-negative integer exponents. They are essential in many areas of mathematics and are known for their versatility and wide applications in solving diverse problems. For example, the equation \(4x^4 + 11x^3 + x^2 + 11x - 3 = 0\) \has a degree of 4, indicating that there could be up to four roots, which can be real or complex. Br> Polynomial equations can be solved using different methods:

• Graphical methods, by plotting the polynomial and analyzing where it crosses the x-axis
• Algebraic methods, which can include factoring for lower-degree equations
• Utilizing the Rational Root Theorem to find potential rational solutions

Understanding how to approach solving polynomial equations is crucial, as it forms the backbone of algebra and paves the way for more advanced studies in mathematics.

Real Roots

Real roots of a polynomial are the x-values where the polynomial equals zero, and these are the points at which the graph of the polynomial crosses or touches the x-axis. Understanding real roots is fundamental, since they often represent practical solutions in real-world applications. For the polynomial \(f(x) = 4x^4 + 11x^3 + x^2 + 11x - 3\), finding real roots involves several steps: Br>

• Start by applying the Rational Root Theorem to identify possible rational values that may be real roots.
• Substitute these candidates into the polynomial equation and check if the result is zero.

Upon testing, you would find that \(x = 1\), \(x = -1\), and \(x = 3\) satisfy \(f(x) = 0\), confirming they are indeed real roots of this particular polynomial. Once found, always verify by substituting back into the original polynomial to ensure accurate results. Understanding these roots allows you to explore and unlock the structural nature of the polynomial function itself.