Problem 41 - Algebra and Trigonometry

Question

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$ P(x)=2 x^{4}-7 x^{3}+3 x^{2}+8 x-4 $$

Step-by-Step Solution

Answer

The rational zeros are 1, 2, and -\(\frac{1}{2}\). Factored form: \((x - 1)(x - 2)^2(2x + 1)\).

Step 1: Identify the Rational Root Theorem

The Rational Root Theorem states that any rational root, in the form of \(\frac{p}{q}\), where \(p\) is a factor of the constant term and \(q\) is a factor of the leading coefficient. For the polynomial \(P(x)=2x^4-7x^3+3x^2+8x-4\), the constant term is \(-4\) and the leading coefficient is \(2\). So, the possible rational roots are \(\pm 1, \pm 2, \pm 4\) divided by \(\pm 1, \pm 2\). This gives \(\pm 1, \pm 2, \pm 4, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{7}{2}, \pm \frac{4}{2}\).

Step 2: Test the Possible Rational Roots

Substitute each possible root into the polynomial to see if it evaluates to zero. By trying these values, we find that \(x = 1\) is indeed a root because \(P(1) = 0\).

Step 3: Factor the Polynomial Using the Found Root

Since \(x = 1\) is a root, \(x - 1\) is a factor of \(P(x)\). Use synthetic division to divide \(P(x)\) by \(x - 1\), resulting in the quotient polynomial \(2x^3 - 5x^2 - 2x + 4\).

Step 4: Repeat the Process for the Quotient

Now, we need to find more roots. Apply the Rational Root Theorem again to the polynomial \(2x^3 - 5x^2 - 2x + 4\). Test the same potential roots. Upon testing, find that \(x = 2\) is another root.

Step 5: Factor Further with the New Root

Since \(x = 2\) is a root of the quotient, \(x - 2\) is a factor. Use synthetic division on \(2x^3 - 5x^2 - 2x + 4\) by \(x - 2\) to get the next quotient polynomial, \(2x^2 - x - 2\).

Step 6: Factor the Quadratic Polynomial

Factor \(2x^2 - x - 2\) as \((2x + 1)(x - 2)\). This gives the remaining factored form along with the linear factors found previously.

Step 7: Final Step: Combine All Factors

Combine all factors to write the fully factored form of the polynomial. Thus, the polynomial can be written as \((x - 1)(x - 2)^2(2x + 1)\).

Key Concepts

Factored Form

When tackling polynomial equations, expressing them in their factored form can be quite enlightening. Factored form involves rewriting the polynomial as a product of its factors, which can include linear factors like \(x - a\) or quadratic factors like \(ax^2 + bx + c\). This form reveals the roots of the polynomial. Each factor corresponds to a root of the polynomial, helping us understand where the polynomial equals zero.

For example, consider the polynomial \(P(x) = 2x^4 - 7x^3 + 3x^2 + 8x - 4\). After applying the Rational Root Theorem and synthetic division, we found that its factored form is \((x - 1)(x - 2)^2(2x + 1)\).

• \(x - 1\) is a linear factor with root \(x = 1\).
• \(x - 2\) repeated, indicating \(x = 2\) is a root with multiplicity 2.
• \(2x + 1\) is another linear factor, leading to the root \(x = -\frac{1}{2}\).

Using factored form helps identify all solutions to \(P(x) = 0\) efficiently.

Synthetic Division

Synthetic division is a simplified form of polynomial division, particularly helpful when dividing by linear factors. This method is quicker and simpler than traditional long division, streamlining calculations and reducing errors.

To use synthetic division, follow these simple steps:

• Write down the coefficients of the polynomial.
• Use the root of the factor \(x - a\) as the divisor 'a'.
• Bring down the leading coefficient to start the process.
• Multiply and add sequentially across the row until you reach the end.

This process results in a quotient polynomial, reducing its degree by one compared to the original.

In the given polynomial \(P(x) = 2x^4 - 7x^3 + 3x^2 + 8x - 4\), after finding \(x = 1\) as a root, synthetic division allowed us to factor out \(x - 1\), resulting in a simpler polynomial \(2x^3 - 5x^2 - 2x + 4\). This quotient can be further analyzed for roots, continuing the factorization process.

Polynomial Roots

Understanding polynomial roots is crucial to solving polynomial equations. These roots are values of \(x\) for which the polynomial evaluates to zero. Finding roots involves identifying potential candidates and confirming them.

The Rational Root Theorem is an invaluable tool for this. It suggests that any rational root \(\frac{p}{q}\) of a polynomial comes from

• \(p\), the factors of the constant term.
• \(q\), the factors of the leading coefficient.

Testing these potential roots allows us to uncover valid solutions.

For \(P(x) = 2x^4 - 7x^3 + 3x^2 + 8x - 4\), initially, we identify and test possible rational roots such as \(\pm 1, \pm 2, \pm \frac{1}{2}\), and find that \(x = 1\) and \(x = 2\) satisfy the equation. Hence, these are actual roots of the polynomial. Once roots are confirmed, they lead us to polynomial factors and, consequently, to the expression in factored form.