Problem 22 - Algebra and Trigonometry

Question

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$ P(x)=x^{3}-4 x^{2}-7 x+10 $$

Step-by-Step Solution

Answer

The rational zeros are \(x = 1, x = 2, x = -5\), and the factored form is \((x - 1)(x - 2)(x + 5)\)."

Step 1: Use the Rational Root Theorem

The Rational Root Theorem states that any rational zero, expressed as \( \frac{p}{q} \), is such that \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient of the polynomial. Here, the constant term is 10 and the leading coefficient is 1. The potential rational zeros are therefore the factors of 10: \( \pm 1, \pm 2, \pm 5, \pm 10 \).

Step 2: Evaluate the Polynomial at Potential Zeros

We evaluate \( P(x) \) at each of the potential zeros found in Step 1: - \( P(1) = 1^3 - 4 \cdot 1^2 - 7 \cdot 1 + 10 = 0 \)- \( P(-1) = (-1)^3 - 4 \cdot (-1)^2 - 7 \cdot (-1) + 10 = 0 \)- Continue testing until you find rational zeros. Thus, \( x = 1 \) is a zero, and \( x = -1 \) is also a zero.

Step 3: Perform Synthetic Division

Use synthetic division to divide \( P(x) \) by \( x - 1 \) to simplify the polynomial: - Coefficients: 1, -4, -7, 10- Synthetic Division gives: \[\begin{array}{c|ccc c}1 & 1 & -4 & -7 & 10 \ & & 1 & -3 & -10 \ \hline & 1 & -3 & -10 & 0 \end{array}\]This leaves us with \( x^2 - 3x - 10 \).

Step 4: Factor the Quotient

We factor the quadratic \( x^2 - 3x - 10 \):- Find two numbers that multiply to \(-10\) and add to \(-3\): \(-5\) and \(2\).- The quadratic factors to \((x - 5)(x + 2)\).

Step 5: Write the Polynomial in Factored Form

Combine the results from synthetic division and further factorization to write the polynomial in its completely factored form:\[ P(x) = (x - 1)(x - 2)(x + 5) \].

Key Concepts

Synthetic Division

Synthetic division is a streamlined method for dividing a polynomial by a linear factor of the form \( x - c \). It simplifies the long division process, especially when dealing with higher-degree polynomials. To perform synthetic division, arrange the coefficients of the polynomial from the highest degree to the constant term.
Next, choose a possible root \( c \) based on the Rational Root Theorem. Place this value outside a division-like box. Write the coefficients inside this box. The process involves repeated addition and multiplication.

• Begin by bringing down the leading coefficient straight to the bottom of the box.
• Multiply this coefficient by the root \( c \) and place it under the next coefficient.
• Add this product to the corresponding coefficient, and write the result directly below.

Repeat these steps until you've gone through all coefficients. If the last value is zero, \( c \) is a root. The remaining numbers represent the coefficients of the divided polynomial. This efficient method is useful when simplifying polynomials and finding zeros.

Polynomial Factoring

Polynomial factoring involves expressing a polynomial as a product of its factors. Factors are polynomials of lower degrees. This technique makes it easier to solve polynomial equations, especially in identifying zeros quickly. When factoring, start by determining if the polynomial can be simplified further using common factors, or special formulas like the difference of squares or perfect square trinomials.
In cases where the polynomial is a quadratic, it can often be factored by finding two numbers that multiply to the product of the leading coefficient and the constant term, while also adding to the linear coefficient. In our example, the quadratic \( x^2 - 3x - 10 \) can be factored by identifying integers that multiply to \(-10\) and add to \(-3\), which are \(-5\) and \(2\). This leads to the factorization \((x - 5)(x + 2)\).
Factoring transforms complex polynomials into simpler components, facilitating easier analysis and solution of polynomial equations.

Rational Zeros

Rational zeros or rational roots of a polynomial are the values of \( x \) at which the polynomial evaluates to zero. According to the Rational Root Theorem, these zeros are likely candidates among the set of fractions \( \frac{p}{q} \), where \( p \) is a factor of the constant term, and \( q \) is a factor of the leading coefficient.
To find rational zeros, start by listing these fractions. Test each candidate by substituting it into the polynomial. If substituting a value yields zero, then that value is a rational zero of the polynomial. This process helps identify simpler regards of the polynomial that can then be used in further simplifying or factoring processes. In mathematical computation, finding rational zeros is an effective means of reducing complex polynomial equations.

Polynomials

Polynomials are mathematical expressions consisting of variables and coefficients. They are written as sums of terms, each term being a product of a constant coefficient and a non-negative power of the variable. The structure of a polynomial, such as \( x^3 - 4x^2 - 7x + 10 \), showcases different degrees, reflecting the power of \( x \) in each term.
Polynomials can be classified based on their degrees:

• Linear: Degree 1, like \( ax + b \).
• Quadratic: Degree 2, like \( ax^2 + bx + c \).
• Cubic: Degree 3, like \( ax^3 + bx^2 + cx + d \).

Understanding polynomials is vital since they're foundational in algebra and calculus. Polynomials allow for modeling various real-world phenomena, providing flexibility in representation due to their operations, such as addition, subtraction, multiplication, division, and factoring.