Problem 11 - Precalculus with Calculus Previews

Question

Verify that each of the indicated numbers are zeros of the given polynomial function \(f\). Find all other zeros and then give the complete factorization of \(f(x)\) $$ -\sqrt{5}, \sqrt{5} ; f(x)=3 x^{4}+x^{3}-17 x^{2}-5 x+10 $$

Step-by-Step Solution

Answer

The complete factorization is \\(f(x) = (x - \sqrt{5})(x + \sqrt{5})(3x^2 + x - 2)\\). The other zeros are the solutions to \\(3x^2 + x - 2 = 0\\), which can be found using the quadratic formula.

Step 1: Verify Given Zeros

Plug each given zero, \(-\sqrt{5}\) and \(\sqrt{5}\), into the polynomial function \(f(x) = 3x^4 + x^3 - 17x^2 - 5x + 10\). Check if the function evaluates to zero for each value.

Step 2: Factor Polynomial Using Given Zeros

Since \(-\sqrt{5}\) and \(\sqrt{5}\) are given zeros, \(x^2 - 5\) is a factor of \(f(x)\). Perform polynomial division of \(f(x)\) by \(x^2 - 5\) to find the quotient polynomial.

Step 3: Perform Polynomial Division

Divide \(3x^4 + x^3 - 17x^2 - 5x + 10\) by \(x^2 - 5\) using polynomial long division or synthetic division to get a quadratic polynomial.

Step 4: Solve Resulting Quadratic Equation

The quotient from the division will be a quadratic polynomial. Solve this quadratic equation using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the remaining zeros.

Step 5: Write Complete Factorization

After finding all zeros (including the given ones), express \(f(x)\) in its completely factored form using the identified zeros.

Key Concepts

Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as a product of its factors. Factors are smaller polynomials that, when multiplied together, give back the original polynomial.
For example, if you have a quadratic polynomial like \(x^2 - 5\), it can be a factor of a larger polynomial \(f(x)\). Knowing that \(-\sqrt{5}\) and \(\sqrt{5}\) are zeros of \(f(x)\), it implies that \(x^2 - 5\) is a factor, since \((x + \sqrt{5})(x - \sqrt{5}) = x^2 - 5\). This way, whenever you have zeros, you can reverse-engineer a factor by thinking of the zeros as solutions to \((x - a)(x - b)\).

• Begin by identifying given zeros of the polynomial.
• Use the zeros to form simple polynomial factors.
• Perform polynomial division to find remaining factors.

Using factorization helps in solving polynomial equations and simplifying expressions.

Quadratic Equation

A quadratic equation is a second-degree polynomial, typically in the form \(ax^2 + bx + c = 0\). Solving quadratic equations is a fundamental skill in algebra, and it involves finding the values of \(x\) for which the equation equals zero.
When tackling a quadratic equation, you have several methods available:

• Factoring: Expressing the quadratic as \((x - p)(x - q) = 0\), meaning \(x = p\) or \(x = q\) are the solutions if they exist.
• Quadratic Formula: For any quadratic \(ax^2 + bx + c\), you can use \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the roots.
• Completing the Square: Altering the equation into a perfect square trinomial.

When you perform a division of a polynomial by a factor and end up with a quotient that is a quadratic expression, applying these methods can easily find the remaining zeros of the original polynomial.

Polynomial Division

Polynomial division is essential in breaking down complex polynomials into simpler, manageable parts. It works similarly to number division but involves variables.
There are different techniques for dividing polynomials:

• Long Division: Divide in a method similar to number division, handling each term individually.
• Synthetic Division: A shortcut used primarily for division by linear polynomials \((x - r)\), this method is quicker and involves less computation.

For the polynomial \(3x^4 + x^3 - 17x^2 - 5x + 10\), dividing by \(x^2 - 5\) helps you simplify it by removing known factors associated with the given zeros, \(-\sqrt{5}\) and \(\sqrt{5}\).
After division, you are often left with a reduced polynomial that is easier to handle, often leading to finding additional zeros or simplifying the equations further.

Polynomial Function

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. It is represented as \(f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\). The degree of the function is determined by the highest power of \(x\).
Polynomials can model various real-world situations, and understanding them can allow for calculating important values known as zeros. Zeros are values of \(x\) for which \(f(x) = 0\). Finding zeros is key for graphing polynomial functions, and often involves factorization, division, and solving equations.

• Zeros might be real or complex numbers.
• Can be used to determine the behavior of the graph.

Each polynomial function can be uniquely factored into linear and irreducible quadratic factors over the set of complex numbers. By finding all the zeros and associated factors, you reveal all the critical aspects of a polynomial function.