Problem 389 - Precalculus

Question

For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor. $$ f(x)=2 x^{3}+x^{2}-5 x+2 ; x+2 $$

Step-by-Step Solution

Answer

The real zeros of \(f(x)\) are \(-2, \frac{1}{2},\) and \(1\).

Step 1: Verify Given Factor

The Factor Theorem states that if \(x + a\) is a factor of \(f(x)\), then \(f(-a) = 0\). Since \(x+2\) is provided as a factor of \(f(x)\), we substitute \(x = -2\) into the polynomial: \(f(-2) = 2(-2)^3 + (-2)^2 - 5(-2) + 2\). This simplifies to: \(-16 + 4 + 10 + 2 = 0\). Since \(f(-2) = 0\), \(x+2\) is confirmed as a factor.

Step 2: Perform Synthetic Division

Using synthetic division, divide \(f(x) = 2x^3 + x^2 - 5x + 2\) by \(x + 2\). Place \(-2\) (the root) on the left and list the coefficients: \(2, 1, -5, 2\). Begin synthetic division: bring down the 2, multiply by \(-2\) (the root), add to next coefficient, and continue the process. The results are: 2, -3, 1, 0, indicating \(f(x) = (x + 2)(2x^2 - 3x + 1)\).

Step 3: Factor the Quadratic

The quotient from synthetic division is \(2x^2 - 3x + 1\). To factor, find two numbers that multiply to \(2 \times 1 = 2\) and add to \(-3\). These numbers are \(-1\) and \(-2\). Factor the quadratic: \(2x^2 - 3x + 1 = (2x - 1)(x - 1)\).

Step 4: Find All Real Zeros

Set each factor of \(f(x) = (x + 2)(2x - 1)(x - 1)\) equal to zero and solve for \(x\):\[x + 2 = 0 \Rightarrow x = -2,\]\[2x - 1 = 0 \Rightarrow x = \frac{1}{2},\]\[x - 1 = 0 \Rightarrow x = 1.\]Thus, the real zeros of \(f(x)\) are \(-2, \frac{1}{2},\) and \(1\).

Key Concepts

Synthetic Division

Synthetic division is a simplified form of polynomial long division, used specifically to divide a polynomial by a binomial of the form \(x + a\). It is a quicker and more efficient method that saves time and reduces calculation errors.

To perform synthetic division:

• Identify the root of the divisor \(x + a\) by rewriting it as \(x = -a\). In our exercise, the divisor is \(x + 2\), so the root is \(-2\).
• List the coefficients of the polynomial from highest degree to lowest. For \(f(x) = 2x^3 + x^2 - 5x + 2\), the coefficients are \(2, 1, -5, 2\).
• Carry out the division by bringing down the leading coefficient, multiply it by the root \(-2\), and add the result to the next coefficient. Repeat these steps until all coefficients have been processed.

The process simplifies the polynomial division, revealing the quotient and confirming the remainder as zero, which indicates \(x + 2\) is actually a factor.

Polynomial Zeros

Polynomial zeros are values of \(x\) that make the polynomial equal to zero. These values are also known as roots or solutions. Finding zeros is crucial in understanding the behavior and graph of the polynomial.

To determine the zeros of a polynomial:

• Use the Factor Theorem, which states that if \(x - a\) is a factor of a polynomial, then \(a\) is a zero.
• Apply synthetic division to simplify the polynomial and continuously break it down into smaller factors.
• Set each factor equal to zero and solve for \(x\).

In the given example, after confirming \(x + 2\) as a factor, we proceed with synthetic division, factorization, and finally finding the zeros: \(x = -2, \frac{1}{2},\) and \(1\). These zeros collectively represent where the polynomial intersects the x-axis.

Quadratic Factorization

Quadratic factorization involves expressing a quadratic expression as the product of two binomials. It's an essential step in solving polynomials that have been reduced to a quadratic form after division.

For the quadratic expression \(2x^2 - 3x + 1\) from our example, we:

• Seek two numbers that multiply to the product of the quadratic's leading coefficient and constant term (e.g., \(2\times 1 = 2\)).
• Find numbers that add to the middle term coefficient (e.g., \(-3\)). These numbers are \(-1\) and \(-2\).
• Reconstruct the quadratic as \((2x - 1)(x - 1)\), indicating it as fully factored.

This factorization simplifies solving for zeros, as each binomial factor set to zero reveals distinct solutions: \(x = \frac{1}{2}\) and \(x = 1\).

Polynomial Functions

Polynomial functions are expressions involving variables with whole number exponents combined via addition, subtraction, or multiplication. They are foundational in algebra and calculus because they model real-world scenarios and abstract mathematical relations.

Key characteristics of polynomial functions:

• The degree of a polynomial denotes the highest exponent and determines its shape and number of zeros.
• Coefficients are numbers representing the multiplicative value of each term.
• They can be factored to uncover zeros, which provide valuable insights into the function's graph.

In our exercise, we dealt with a third-degree polynomial \(f(x) = 2x^3 + x^2 - 5x + 2\), reflecting three zeros based on the degree. Understanding how polynomial functions work aids in predicting and solving complex algebraic relationships efficiently.