Problem 16 - College Algebra with Corequisite Support

Question

For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor. $$ f(x)=3 x^{3}+x^{2}-20 x+12 ; \quad x+3 $$

Step-by-Step Solution

Answer

The real zeros of the polynomial are \(-3, 2,\) and \(\frac{2}{3}.\)

Step 1: Use the Factor Theorem

The Factor Theorem tells us that if \( x + a \) is a factor of the polynomial \( f(x) \), then \( f(-a) = 0 \). Here, the given factor is \( x + 3 \), so substitute \(-3\) into \( f(x) \): \[ f(-3) = 3(-3)^3 + (-3)^2 - 20(-3) + 12. \]

Step 2: Evaluate the Polynomial at \( x = -3 \)

Compute \( f(-3) \):\[ f(-3) = 3(-27) + 9 + 60 + 12 \]\[ f(-3) = -81 + 9 + 60 + 12 \]\[ f(-3) = 0. \]Since \( f(-3) = 0 \), \( x + 3 \) is indeed a factor.

Step 3: Perform Synthetic Division

Use synthetic division with divisor \( x + 3 \) (or root \( -3 \)) to divide the polynomial:1. Write the coefficients of \( f(x) \): 3, 1, -20, 12.2. Bring down the leading coefficient (3).3. Multiply the root (-3) with the number just written (3) and write the product under the next coefficient (1).4. Add the numbers: \( 1 + (-9) = -8 \).5. Repeat this process: multiply \(-3\) with \(-8\) (write the product under \(-20\)), add: \(-20 + 24 = 4\).6. Multiply \(-3\) with 4, write the product under 12, add: \(12 - 12 = 0\).The result of the division is the quadratic \( 3x^2 - 8x + 4 \).

Step 4: Solve the Quadratic Equation

The quadratic equation obtained is \( 3x^2 - 8x + 4 = 0 \). Use the Quadratic Formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):- Here, \( a = 3 \), \( b = -8 \), \( c = 4 \).- Compute the discriminant: \( b^2 - 4ac = (-8)^2 - 4(3)(4) = 64 - 48 = 16 \).- Solve using the formula: \[ x = \frac{-(-8) \pm \sqrt{16}}{2 \cdot 3} = \frac{8 \pm 4}{6}. \]- The solutions are \( x = \frac{8+4}{6} = 2 \) and \( x = \frac{8-4}{6} = \frac{2}{3}. \)

Step 5: Compile the Real Zeros

From the previous steps, we confirmed \( x = -3 \) as a zero, and from solving the quadratic, we found the additional zeros: \( x = 2 \) and \( x = \frac{2}{3} \).

Key Concepts

Polynomial Functions

Polynomial functions are expressions that involve variables raised to whole number exponents. The general form of a polynomial function is:

• The highest power of the variable is known as the degree.
• Polynomial functions can have varying degrees, such as linear (degree 1), quadratic (degree 2), cubic (degree 3), and so on.
• Each term has a coefficient and a power of the variable.

The degree of a polynomial is significant because it tells us about the number of potential solutions \( (x) \). In the given exercise, the polynomial function is cubic: \[ f(x) = 3x^3 + x^2 - 20x + 12 \] This means it might have up to three real solutions because of its degree. Polynomial functions are continuous and smooth curves with shapes that vary depending on the degree and the leading coefficient.

Synthetic Division

Synthetic division is a simplified method of dividing a polynomial by a divisor of the form \(x + a\). It is much faster than long division and particularly helpful for finding factors and zeros. When applying synthetic division:

• Use only the coefficients of the polynomial.
• Perform the calculation with the opposite sign of the divisor’s term.
• The process produces a new set of coefficients representing the quotient polynomial.

In the step-by-step solution provided, synthetic division was used to divide the original polynomial by \( x + 3 \). The root or zero used was \(-3\), the opposite of the factor given. The result was the quadratic expression:\[ 3x^2 - 8x + 4 \] This expression indicates the polynomial after removing the factor \( x + 3 \). With synthetic division, we efficiently reduced the degree of the polynomial, allowing us to analyze simpler forms of the function.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation of the form \( ax^2 + bx + c = 0 \). Quadratics often feature prominently after using factorization techniques like synthetic division. Solving these equations can provide the remaining real zeros of a polynomial function:

• The quadratic formula:\[x= \frac{-b \pm \sqrt{b^2 -4ac}}{2a}\]
• Allows us to find the roots of any quadratic equation.

Applying this to the quadratic \(3x^2 - 8x + 4 = 0\), identified through synthetic division, we discovered the remaining real zeros:

• Compute the discriminant:
• \(b^2 - 4ac = 16\)
• Ensure it is non-negative to assure real roots.

As we computed before, the solutions are \(x = 2\) and \(x = \frac{2}{3}\). Quadratics not only help in identifying solutions but also in analyzing the characteristics of polynomial graphs.

Real Zeros

Real zeros of a polynomial are the solutions where the polynomial equals zero. They are crucial because they represent the points where the graph of the polynomial intersects the x-axis:

• Using the Factor Theorem helps in testing potential zeros.
• Synthetic division further helps verify these zeros and simplifies the polynomial for further solving.

In the exercise, by finding \( f(-3) = 0 \) we confirmed \(-3\) is a real zero.

• The subsequent quadratic equation \(3x^2 - 8x + 4 = 0\) provided additional zeros after computation.
• Zeros found: \(-3, 2, \frac{2}{3}\)

These zeros are the x-coordinates where the function value is zero. Understanding real zeros enables students to sketch the graph of the function and analyze the behavior of polynomial functions.