Problem 25 - Precalculus : Building Concepts and Connections

Question

One zero of each polynomial is given. Use it to express the polynomial as a product of linear and irreducible quadratic factors. $$2 x^{3}-9 x^{2}-11 x+30 ; \text { zero: } x=5$$

Step-by-Step Solution

Answer

The polynomial \(2x^{3}-9x^{2}-11x+30\) can be expressed as a product of linear and irreducible quadratic factors as \((x - 5)(2x^2 + x - 6)\), where \((x - 5)\) comes from our initial root and \((2x^2 + x - 6)\) the factored form of the quadratic polynomial gotten from the synthetic division process.

Step 1: Synthetic Division

The first step is to perform synthetic division using the given zero \(x = 5\). Lay out the synthetic division tableau. The coefficients of the polynomial \(2x^3-9x^2-11x+30\) go on the top, while the zero \(x=5\) goes on the left side. Following synthetic division processes, drag down the leading coefficient, multiply it by 5 and write the result under the second coefficient, add the numbers in the second column, multiply the result by 5, and so on until every coefficient has been used.

Step 2: Construct Reduced Polynomial

From the synthetic division process, the numbers at the bottom row represent the coefficients of the reduced polynomial. Since we started with a cubic polynomial and used one real root, this will give us a quadratic polynomial. Write this polynomial using the coefficients from the bottom row of the synthetic division tableau.

Step 3: Factoring the Quadratic Polynomial

Factorize the resulting quadratic polynomial to get the remaining roots and thus the irreducible quadratic factors. If the quadratic doesn't factorize over the integers, it is already an irreducible quadratic factor.

Step 4: Finalize the Answer

Combine the linear factor from the provided zero with the obtained quadratic factors to express the original polynomial as a product of factors.

Key Concepts

Polynomial Factoring

Polynomial factoring is a method used to express a polynomial as a product of its simpler component parts, which are called factors. This is useful because it can help us solve polynomial equations by finding their roots. In our exercise, we have a polynomial \(2x^3 - 9x^2 - 11x + 30\), and we know one of its roots, \(x = 5\). Knowing a root helps us because we can use it to break down the polynomial using synthetic division. This step simplifies the complex cubic polynomial into a more manageable quadratic polynomial. Once reduced, if this quadratic polynomial can be factored further, we split it into linear or irreducible quadratic factors. This process lets us express the original polynomial as a product of simpler polynomials, which shows all the roots distinctly.

• Start with the given polynomial.
• Use a known root to simplify via synthetic division.
• Factor the remaining polynomial further if possible.

Factoring helps not just in solving equations but also in simplifying expressions in calculus, algebra, and beyond.

Cubic Polynomials

Cubic polynomials are polynomial expressions in which the highest degree of the variable is three. They generally take the form \(ax^3 + bx^2 + cx + d\). In our exercise, the polynomial is \(2x^3 - 9x^2 - 11x + 30\). Solving cubic polynomials involves finding their roots, which are the values of \(x\) where the polynomial equals zero.

• They usually have one to three real roots.
• The degree of the polynomial tells us how many total roots there are, counting multiplicity.
• Knowing even one root can help break down the polynomial simpler terms.

Cubic polynomials can be tricky, but with systematic techniques like synthetic division and factoring, solving them becomes manageable. They often appear in various mathematical models, making understanding their breakdown essential.

Polynomial Roots

Polynomial roots, also known as zeros, are the solutions to the equation formed by setting a polynomial equal to zero. They represent the values of \(x\) where the polynomial crosses the x-axis on a graph. For our cubic polynomial \(2x^3 - 9x^2 - 11x + 30\), the given root is \(x = 5\). Once a root is known, it can be used to perform synthetic division, simplifying the polynomial to discover additional roots.

• Roots can be real or complex numbers.
• Knowing a single root allows us to reduce the polynomial's degree.
• Linear roots are derived easily, while quadratic factoring may reveal other roots.

Identifying all roots is crucial for understanding the complete behavior of polynomial functions, especially in graphing and modeling real-world situations. They represent the critical points and provide insights into the function's intersections and turning points.