Problem 17 - 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. $$ f(x)=2 x^{3}+3 x^{2}+x+6 ; \quad x+2 $$

Step-by-Step Solution

Answer

The only real zero is \( x = -2 \).

Step 1: Apply the Factor Theorem

The Factor Theorem states that if \( x + 2 \) is a factor of the polynomial \( f(x) = 2x^3 + 3x^2 + x + 6 \), then substituting \( x = -2 \) into \( f(x) \) should result in 0. Calculate \( f(-2) \) to verify the factor.\[f(-2) = 2(-2)^3 + 3(-2)^2 + (-2) + 6 = -16 + 12 - 2 + 6 = 0.\]Since the result is 0, \( x + 2 \) is indeed a factor.

Step 2: Perform Synthetic Division

Use synthetic division to divide the polynomial \( 2x^3 + 3x^2 + x + 6 \) by \( x + 2 \). Carry out the division with \( -2 \) as the root. Write 2, 3, 1, and 6 as coefficients and perform the division steps.- Bring down the first coefficient: 2.- Multiply -2 by 2, add to the second coefficient: 3 + (-4) = -1.- Multiply -2 by -1, add to the third coefficient: 1 + 2 = 3.- Multiply -2 by 3, add to the fourth coefficient: 6 + (-6) = 0.The quotient is \( 2x^2 - x + 3 \) with a remainder of 0.

Step 3: Find Zeros of the Quotient Polynomial

We need to find the zeros of the quotient polynomial \( 2x^2 - x + 3 \). Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 2 \), \( b = -1 \), and \( c = 3 \). Calculate the discriminant:\[b^2 - 4ac = (-1)^2 - 4(2)(3) = 1 - 24 = -23\]Since the discriminant is negative, there are no real zeros. This polynomial does not yield additional real zeros.

Key Concepts

Real Zeros of Polynomial

When dealing with polynomials, real zeros are the points where the polynomial equals zero. Finding these zeros is crucial in understanding the behavior of the polynomial's graph. Knowing the zeros helps you determine the x-intercepts of the graph, showing you where the graph crosses the x-axis. For the polynomial function given in the exercise, we use the Factor Theorem. This theorem states that if a polynomial \( f(x) \) has a factor \( (x - r) \), then \( f(r) = 0 \). Applying this, we verify whether \( x+2 \) is a factor by substituting \( x = -2 \) into the polynomial.

• If the result is zero, then \( x+2 \) is indeed a factor, and \( -2 \) is a real zero of the polynomial.
• In our exercise, we found \( f(-2) = 0 \), confirming \( x = -2 \) as a real zero.

Understanding this concept helps demystify how factors and zeros are intertwined in polynomial equations.

Synthetic Division

Synthetic division is a shortcut for dividing a polynomial by a binomial of the form \( x - c \). It's simpler than long division and is especially useful when using the Factor Theorem. In the exercise, we apply synthetic division to divide \( 2x^3 + 3x^2 + x + 6 \) by \( x + 2 \), which corresponds to a root \( -2 \). Here's how you can perform this division step-by-step:

• Write down the coefficients of the polynomial: 2, 3, 1, 6.
• Use \( -2 \) as the divisor (since \( x+2 \) makes \( x = -2 \)).
• Bring down the leading coefficient, 2.
• Multiply \(-2\) by 2 and add it to the next coefficient 3, giving you -1.
• Continue this process until all coefficients have been used, verifying a zero remainder.

This results in the quotient \( 2x^2 - x + 3 \) with a remainder of zero, confirming \( x+2 \) is a factor of the polynomial. This method is efficient in reducing the polynomial's degree while maintaining accuracy.

Quadratic Formula

The quadratic formula is a powerful tool for finding the zeros of a quadratic polynomial, expressed as \( ax^2 + bx + c = 0 \). Even when you can't factor the quadratic easily, the formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) provides a reliable solution.In the exercise, after using synthetic division, we are left with the quadratic \( 2x^2 - x + 3 \). Plugging into the quadratic formula:

• We have \( a = 2 \), \( b = -1 \), and \( c = 3 \).
• Calculate the discriminant \( b^2 - 4ac \). This gives \((-1)^2 - 4(2)(3) = 1 - 24 = -23\).
• A negative discriminant indicates that the quadratic has no real zeros, as square roots of negative numbers are not real.

In this context, the quadratic formula confirms that there are no additional real zeros to the polynomial. This highlights the importance of the discriminant in determining the nature of the roots.