Problem 11 - Precalculus with Calculus Previews

Question

Find all rational zeros of the given polynomial function \(f\). $$ f(x)=x^{4}+6 x^{3}-7 x $$

Step-by-Step Solution

Answer

The rational zeros are 0, 1, and -7.

Step 1: Identify Possible Rational Zeros

To identify the possible rational zeros of the polynomial \( f(x) = x^4 + 6x^3 - 7x \), we use the Rational Root Theorem. This theorem states that any rational zero, \( \frac{p}{q} \), where \( p \) is a factor of the constant term and \( q \) is a factor of the leading coefficient, must be a factor of the constant term (which is 0 here) and the leading coefficient (which is 1). Hence, the only possible rational zero here is \( x = 0 \).

Step 2: Evaluate the Polynomial at the Possible Zero

Substitute \( x = 0 \) into the polynomial to verify if it is indeed a zero. Calculate: \[ f(0) = 0^4 + 6 \cdot 0^3 - 7 \cdot 0 = 0. \] Since \( f(0) = 0 \), \( x = 0 \) is a zero of the polynomial.

Step 3: Perform Polynomial Division

Now that we have identified \( x = 0 \) as a zero, perform polynomial division to factor out \( x \) from \( f(x) = x^4 + 6x^3 - 7x \). Divide the polynomial by \( x \) to obtain: \[ f(x) = x(x^3 + 6x^2 - 7). \]

Step 4: Solve the Remaining Polynomial

Solve for zeros of the remaining polynomial \( x^3 + 6x^2 - 7 \). Notice that this cubic polynomial does not have a constant term, so we can start by factoring out \( x \) again: \[ x^3 + 6x^2 - 7 = x(x^2 + 6x - 7). \] Find the zeros of the quadratic \( x^2 + 6x - 7 \) using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \] where \( a = 1, b = 6, c = -7 \).

Step 5: Compute the Zeros Using the Quadratic Formula

Substitute \( a = 1, b = 6, c = -7 \) into the quadratic formula: \[ x = \frac{-6 \pm \sqrt{6^2 - 4 \times 1 \times (-7)}}{2 \times 1} = \frac{-6 \pm \sqrt{36 + 28}}{2}. \] Simplify further: \[ x = \frac{-6 \pm \sqrt{64}}{2} = \frac{-6 \pm 8}{2}. \] Thus, \( x = 1 \) and \( x = -7 \).

Step 6: Collect All Rational Zeros

All factors of \( f(x) = x^4 + 6x^3 - 7x \) are \( x = 0, x = 1, \) and \( x = -7 \). Verify that these zeros satisfy the polynomial. Therefore, the rational zeros are 0, 1, and -7.

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a powerful tool for identifying potential rational zeros of a polynomial. The theorem provides us with a list of possible candidates that we can test directly to see if they are actual zeros. The formula states that if a polynomial has rational zeros of the form \(\frac{p}{q}\), \( p \) should be a factor of the constant term, and \( q \) should be a factor of the leading coefficient.

In this particular example, the polynomial is \( f(x) = x^4 + 6x^3 - 7x \). Here, the constant term at the end of the polynomial is actually zero, and the leading coefficient is 1.
If we only have one potential zero when applying the theorem, it suggests that zero is a rational zero, which is indeed the case here.

This simplification means the theorem points directly to \( x=0 \) as the only rational potential zero, simplifying the identification process.
This insight helps efficiently identify zeros without the need for guesswork or complex calculations.

Polynomial Division

Polynomial division works similarly to regular division with numbers, but instead of numbers, we work with polynomials. When you perform polynomial division, you divide one polynomial by another, resulting in a quotient and possibly a remainder. This method is invaluable when simplifying polynomials and finding factors.

In the context of the exercise, once we've identified \( x = 0 \) as a zero of the polynomial \( f(x) = x^4 + 6x^3 - 7x \), we use polynomial division to factor out \(x\) from the polynomial. By doing so, the polynomial simplifies to \( x(x^3 + 6x^2 - 7) \).

• The division allows us to break down the initial polynomial into more manageable components, making it easier to work with.
• This step effectively reduces the degree of the polynomial, which is crucial when we need to solve for additional zeros in subsequent steps.

Since polynomials can often be complex and intimidating, this breakdown through division simplifies the expressions and points us directly to solutions like further factoring or using the quadratic formula on what remains.

Quadratic Formula

The quadratic formula is a universal tool for finding zeros of quadratic polynomials, expressed as \( ax^2 + bx + c = 0 \). This formula is particularly helpful when factoring is difficult or impossible.

The formula itself is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] With these values:

• \( a \) stands for the coefficient of \( x^2 \)
• \( b \) is the coefficient of \( x \)
• \( c \) is the constant term

In the step-by-step solution, after dividing the original polynomial, we encounter a remainder polynomial \( x^2 + 6x - 7 \). To find the zeros of this quadratic, we substitute \( a = 1 \), \( b = 6 \), and \( c = -7 \) into the quadratic formula to solve:
\[ x = \frac{-6 \pm \sqrt{6^2 - 4 \times 1 \times (-7)}}{2 \times 1} \]
Simplifying, it yields two solutions: \( x = 1 \) and \( x = -7 \).

This approach confirms the zeros and is especially critical when factoring by simpler means isn't immediately apparent, providing us a concrete method to arrive at all solutions systematically.