Problem 28 - Precalculus : Building Concepts and Connections

Question

Find all the real zeros of the polynomial. $$P(t)=6 t^{3}-4 t^{2}+3 t-2$$

Step-by-Step Solution

Answer

After the steps, your solutions (real numbers) are the valid roots you found.

Step 1: Apply Rational Root Theorem

First, list all the possible rational roots of the polynomial using the Rational Root Theorem. These are the ratio of the factors of constant term to the factors of the leading coefficient.

Step 2: Evaluate potential roots

Next, evaluate the polynomial at each potential rational root until you find one that makes the polynomial equals zero. That's a valid solution.

Step 3: Simplify the polynomial

Now, use long division or synthetic division of the polynomial by \((t - \text{{root}})\) to simplify the polynomial to a quadratic polynomial.

Step 4: Quadratic Formula

Finally, use the quadratic formula \(-b±√(b²-4ac)/2a\) to solve for the roots of the quadratic polynomial. These will be the remaining solutions to the original cubic polynomial.

Key Concepts

Rational Root Theorem

The Rational Root Theorem is an essential tool for solving polynomial equations, especially when it comes to finding the real zeros of a polynomial like \( P(t) = 6t^3 - 4t^2 + 3t - 2 \). This theorem provides a systematic way to identify all possible rational zeros of a polynomial equation.

According to the theorem, if a polynomial has rational zeros, they must be in the form of a fraction \( \frac{p}{q} \), where \( p \) is a factor of the constant term (here, -2) and \( q \) is a factor of the leading coefficient (here, 6). These factors include both positive and negative values. For the given polynomial, the possible factors of -2 are ±1, ±2, and the factors of 6 are ±1, ±2, ±3, ±6. This yields the potential rational roots as ±1, ±1/2, ±1/3, ±1/6, ±2, ±2/3.

Upon testing these possible roots, if we find that plugging a root into the polynomial yields zero, we have discovered a real zero of the polynomial. It's efficient to test these potential roots before moving on to other methods since it can greatly simplify the equation.

Synthetic Division

Synthetic division is preferred over long division when dividing a polynomial by a binomial of the form \( (t-\text{root}) \) because it is less cumbersome and allows for quicker simplification. It’s especially useful after using the Rational Root Theorem to identify a real zero since it helps reduce the polynomial to a lower degree.

Here is a simplified step-by-step process:

• Write down the coefficients of the polynomial.
• Bring down the leading coefficient to the bottom row.
• Multiply this coefficient by the root and write the result under the next coefficient.
• Add the numbers in the column and write the result to the bottom row.
• Repeat the multiplication and addition process until every coefficient has been used.

The numbers on the bottom row give the coefficients of the quotient polynomial. For example, after identifying a real zero \( r \), we can use synthetic division to divide the polynomial \( P(t) \) by \( (t – r) \), which will result in a quadratic (degree 2) polynomial that is easier to solve.

Quadratic Formula

Once a cubic polynomial is simplified to a quadratic using methods like synthetic division, we can utilize the quadratic formula to determine the remaining real zeros. The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation of the form \( ax^2 + bx + c = 0 \).

The quadratic formula provides the solutions to any quadratic equation, accounting for all possibilities including two real roots, one repeated real root (in the case of a perfect square trinomial), or two complex roots (when the discriminant, \( b^2 - 4ac \), is negative).

For the given problem, once we’ve used the earlier methods to reduce the original cubic polynomial to a quadratic, we’ll be able to identify the last two of the real zeros by substituting the coefficients of this new quadratic equation into the formula. If the discriminant is positive, we get two distinct real zeros; if it’s zero, we get a single real zero with multiplicity two.