Problem 51 - Algebra and Trigonometry

Question

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example \(3(\mathrm{a}) .\) $$ P(x)=x^{4}-7 x^{3}+14 x^{2}-3 x-9 $$

Step-by-Step Solution

Answer

The real zeros are \( x = 3 \) and possibly repeated cases or other found through trials if direct steps align simplification beyond trial.

Step 1: Evaluate Polynomial for Rational Zeros

First, use the Rational Root Theorem to check potential rational zeros, which are factors of the constant term, -9, divided by factors of the leading coefficient, 1. Hence, the potential rational zeros could be \( \pm 1, \pm 3, \pm 9 \). Evaluate \( P(x) \) for these values to check if any of them are roots.

Step 2: Check \( P(1) \)

Calculate \( P(1) = (1)^4 - 7(1)^3 + 14(1)^2 - 3(1) - 9 = 1 - 7 + 14 - 3 - 9 = -4 \). So, \( x = 1 \) is not a root.

Step 3: Check \( P(-1) \)

Compute \( P(-1) = (-1)^4 - 7(-1)^3 + 14(-1)^2 - 3(-1) - 9 = 1 + 7 + 14 + 3 - 9 = 16 \). Thus, \( x = -1 \) is not a root.

Step 4: Trial and Error for Rational Roots

Continue checking the remaining potential rational zeros: \( x = 3 \) and \( x = -3 \). Compute: \( P(3) = (3)^4 - 7(3)^3 + 14(3)^2 - 3(3) - 9 = 81 - 189 + 126 - 9 - 9 = 0 \). So, \( x = 3 \) is a root.

Step 5: Factor the Polynomial Using the Found Root

Since \( x = 3 \) is a root, divide \( P(x) \) by \( x - 3 \) using synthetic division or polynomial division.

Step 6: Synthetic Division with \( x - 3 \)

Perform synthetic division: \( 1 | -7 | 14 | -3 | -9 \) with \( 3 \). The result gives the quotient polynomial \( x^3 - 4x^2 + 2x - 3 \).

Step 7: Solve the Quotient Polynomial

Check for rational roots or use the quadratic formula for further roots of \( x^3 - 4x^2 + 2x - 3 \). Notice that for \( \text{remainder zero from synthetic} \) means \( x = 3 \) is a repeated root.

Step 8: Analyze \( x^3 - 4x^2 + 2x - 3 \)

Factorize further to find more zeros: By trial \( P(1) \). Check if \( x = 1 \) or such further rational trial can solve this.

Step 9: Further Division if Needed

Using \( x - 1 \), check by dividing: conduct trial for further polynomial in division if proceeding to simpler. Use substitution for poly check for irrational.

Key Concepts

Rational Root Theorem

The Rational Root Theorem is a very handy tool when you want to find potential rational zeros of a polynomial. It tells us that any rational solution of the polynomial equation formed by factors of the constant term and factors of the leading coefficient can give possible roots.

• Take the constant term (-9 in this case) and list all its factors: \( \pm 1, \pm 3, \pm 9 \).
• Then, take the leading coefficient, which is 1, and list its factors: \( \pm 1 \).
• Now, form possible rational roots by creating fractions from these factors. However, since our leading coefficient is 1, the possible roots remain the same as the factors of the constant value: \( \pm 1, \pm 3, \pm 9 \).

This theorem greatly narrows down the number of roots you need to test when tackling polynomial equations.

Synthetic Division

Synthetic division is a simplified form of polynomial division that only works when dividing by linear factors and is much quicker and less cumbersome than long division.

• Here, we set up a synthetic division framework using the found root. For example, we have a root 3, so we divide \( P(x) \) by \( x - 3 \).
• Write down the coefficients of the polynomial: \( 1, -7, 14, -3, -9 \).
• Perform the division by bringing down the first coefficient, multiplying it by the root, adding the result to the next coefficient, and continuing in this manner until completion.

This process helps us determine the quotient polynomial, revealing further insights into the polynomial's structure.

Quadratic Formula

When reducing a polynomial down to a quadratic equation, the quadratic formula becomes very useful for solving it. The formula \( ax^2 + bx + c = 0 \) can be solved using:

• The quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
• The discriminant \( b^2-4ac \) is under the square root. It tells us about the nature of the roots - whether they are real, repeated, or complex.
• In a situation where the quotient polynomial is quadratic, use this formula to find exact roots.

Employing the quadratic formula becomes essential when other methods like factoring or trial and error are not feasible.

Polynomial Division

Polynomial division is the process of dividing two polynomials, akin to how you would divide numbers. It helps break down complex polynomials into more manageable pieces.

• When you identify a root, say \( x = 3 \), it indicates that \( x - 3 \) is a factor. You would then perform polynomial division to see what remains of the polynomial.
• This could involve long division in cases where synthetic division is not applicable, especially with non-linear factors.
• The division helps determine if there are any additional roots to be factored out from the remaining polynomial.

Polynomial division and synthetic division often work hand-in-hand, aiding significantly in finding all real zeros of a polynomial.