Problem 16 - Algebra 2

Question

Find all of the rational zeros of each function. \(f(x)=x^{3}+x^{2}-80 x-300\)

Step-by-Step Solution

Answer

The rational zeros are 5 and -10.

Step 1: Identify Possible Rational Zeros

Apply the Rational Root Theorem, which states that any rational zero of the polynomial \( f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \) is of the form \( \frac{p}{q} \), where \( p \) divides the constant term \( a_0 \) and \( q \) divides the leading coefficient \( a_n \). For \( f(x) = x^3 + x^2 - 80x - 300 \), \( a_0 = -300 \) and \( a_n = 1 \). Hence, the possible rational zeros are the divisors of -300, which are \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 10, \pm 12, \pm 15, \pm 20, \pm 25, \pm 30, \pm 50, \pm 60, \pm 75, \pm 100, \pm 150, \pm 300 \).

Step 2: Test Possible Rational Zeros

Substitute possible rational zeros into the polynomial to determine which ones are actual zeros. This can be done by direct substitution or by using synthetic division.

Step 3: Verify a Zero

Using synthetic division with \( x = 5 \), we perform:| 5 | 1 | 1 | -80 | -300 ||--------|-------|-------|--------|--------|| ↓ | | 5 | 30 | -250 || | 1 | 6 | -50 | 0 |The remainder is 0, so 5 is a zero of \( f(x) \).

Step 4: Solve the Quotient Polynomial

After using synthetic division, the quotient polynomial is \( x^2 + 6x - 50 \). Factor or use the quadratic formula to find the remaining zeros if factorization is not readily apparent.

Step 5: Factorize the Quadratic Polynomial

The quadratic polynomial \( x^2 + 6x - 50 \) can be factorized. By finding two numbers that multiply to -50 and add to 6, we find (x + 10)(x - 5) = 0.

Step 6: Solve the Factorized Equation

Set each factor equal to zero: 1. \( x + 10 = 0 \) yields \( x = -10 \).2. \( x - 5 = 0 \) yields \( x = 5 \).

Step 7: List All Rational Zeros

The rational zeros of \( f(x) \) are the numbers identified in the previous steps: \( x = 5 \) (with a multiplicity of 2, but listed once as a rational zero) and \( x = -10 \).

Key Concepts

Polynomial functions

A polynomial function is an expression that involves variables raised to positive integer powers and coefficients. It's like building blocks of algebra that combine different terms. For example, a simple polynomial might look like this: \( x^2 + 3x + 2 \). In general, a polynomial is usually written as \( f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0 \), where each \( a \) represents a coefficient, and \( n \) is the degree of the polynomial, indicating its highest power.Polynomials can have different shapes depending on their degrees. Quadratic polynomials (like \( x^2 + 3x + 2 \)) form parabolas, while cubic polynomials (like \( x^3 \)) have a more curved, snake-like shape. Understanding polynomial functions is fundamental when seeking their roots or zeros, as it allows us to explore different mathematical techniques, such as the Rational Root Theorem, to find solutions.

Synthetic division

Synthetic division is a streamlined method of dividing a polynomial by a linear divisor of the form \( x - c \). This method is especially handy because it simplifies the long division process.To use synthetic division, you follow these steps:1. Write down the coefficients of the polynomial.2. Use the zero of the divisor (i.e., if the divisor is \( x - 3 \), use 3) on the left.3. Bring down the first coefficient to the bottom row.4. Multiply this coefficient by the zero and add this value to the next coefficient.5. Repeat this process until you work through all coefficients.The bottom row will provide the coefficients of the quotient polynomial, while any remainder appears in the last column. When the remainder is zero, it shows that the divisor is indeed a root of the original polynomial.

Quadratic factoring

Quadratic factoring involves breaking down a quadratic expression into the product of simpler expressions, or factors. This method is useful for solving quadratic equations that can be expressed in the form \( ax^2 + bx + c \).Here's how to factor a quadratic:- Look for two numbers that multiply to the constant term (\( c \)) and add to the coefficient of the middle term (\( b \)).- Once you find these numbers, write the quadratic as a product of two linear binomials. For example, \( x^2 + 5x + 6 \) factors to \( (x + 2)(x + 3) \), because 2 times 3 equals 6, and 2 plus 3 equals 5.This method can be applied after synthetic division when the polynomial has been reduced to a quadratic form. It allows us to find the remaining zeros of the polynomial function.

Rational zeros

Rational zeros are the solutions to a polynomial equation that can be expressed as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers. To find these zeros, we often use the Rational Root Theorem.The Rational Root Theorem posits that any potential rational zero of the polynomial \( f(x) = a_nx^n + ... + a_0 \) will be of the form \( \frac{p}{q} \), where \( p \) divides the constant term \( a_0 \), and \( q \) divides the leading coefficient \( a_n \).To identify rational zeros:

• List all the possible values of \( \frac{p}{q} \) using the divisors of the constant term and the leading coefficient.
• Test these potential zeros using substitution or synthetic division to find the actual zeros.

Once rational zeros are verified, they can provide crucial checkpoints for factoring polynomials and solving equations efficiently.