Problem 35 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

Zeros: \(x = 1, \frac{1}{2}, -\frac{3}{2}\). Factored form: \((x - 1)(2x - 1)(2x + 3)\).

Step 1: Understand the Problem

We need to find all rational zeros of the polynomial \(P(x) = 4x^3 - 7x + 3\). After finding the zeros, we'll express the polynomial in its factored form using these zeros.

Step 2: Determine Possible Rational Zeros

Use the Rational Root Theorem, which states that any rational zero, expressed as \(\frac{p}{q}\), has \(p\) as a factor of the constant term (3) and \(q\) as a factor of the leading coefficient (4). The possible rational zeros are the factors of \(\frac{3}{4}\): \(\pm 1, \pm 3, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{1}{4}, \pm \frac{3}{4}\).

Step 3: Test Possible Zeros

Use synthetic division or direct substitution to test each possible zero. Start with \(x = 1\). Substituting \(x = 1\) into the polynomial gives \(4(1)^3 - 7(1) + 3 = 0\). Thus, \(x = 1\) is a zero.

Step 4: Factor the Polynomial Given One Zero

With \(x - 1\) as a factor, perform synthetic division of \(4x^3 - 7x + 3\) by \(x - 1\). The division yields \(4x^2 + 4x - 3\).

Step 5: Find Zeros of the Quotient Quadratic

To find the remaining zeros, solve \(4x^2 + 4x - 3 = 0\) using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 4\), \(b = 4\), and \(c = -3\). Substitute these values to get \(x = \frac{-4 \pm \sqrt{16 + 48}}{8} = \frac{-4 \pm \sqrt{64}}{8} = \frac{-4 \pm 8}{8}\). The solutions are \(x = \frac{1}{2}\) and \(x = -\frac{3}{2}\).

Step 6: Write the Polynomial in Factored Form

Using the zeros found, the polynomial in factored form is \(P(x) = 4(x - 1)(x - \frac{1}{2})(x + \frac{3}{2})\). Simplify the factors to write it as \(P(x) = (x - 1)(2x - 1)(2x + 3)\).

Step 7: Verify the Factored Form

Expand \((x - 1)(2x - 1)(2x + 3)\) to ensure it matches the original polynomial. You will find that it equals \(4x^3 - 7x + 3\), confirming the factors are correct.

Key Concepts

Synthetic Division

Synthetic division is a streamlined version of the long division process specifically for polynomials. It's a quick way to test potential rational roots derived from the Rational Root Theorem. To begin, arrange the polynomial coefficients in order. For instance, with the polynomial \(4x^3 - 7x + 3\), you use the coefficients \([4, 0, -7, 3]\). Notice the zero for the \(x^2\) term since it is missing.
Next, choose a potential root, say \(x = 1\), and place it outside the division. The first coefficient, 4, is brought down. Multiply 4 by the potential root 1 to get 4, placing it under the next coefficient. Add, then repeat the multiply-add cycle.
The process ends when all coefficients are used. A remainder of zero confirms that the tested value (1 in this case) is indeed a root.

• Always use coefficients from highest to lowest degree. Substitute placeholders if necessary to maintain the polynomial structure.
• A remainder of zero indicates the test number is a root.
• If the remainder isn't zero, that number isn't a root, and you move to the next possibility.

Quadratic Formula

The quadratic formula is an essential tool for solving quadratic equations of form \(ax^2 + bx + c = 0\). When synthetic division simplifies a cubic polynomial to a quadratic, this formula helps find additional roots. In our exercise, after performing synthetic division, we solve the quotient \(4x^2 + 4x - 3 = 0\) using the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\\]Plug in the specific values: \(a = 4\), \(b = 4\), and \(c = -3\). The steps are as follows:

• Calculate the discriminant: \(b^2 - 4ac\). Here it becomes \(16 + 48 = 64\).
• The square root of 64 is 8, simplifying the expression to \(x = \frac{-4 \pm 8}{8}\).
• This results in solutions: \(x = \frac{1}{2}\) and \(x = -\frac{3}{2}\).

These roots match our polynomial's simplified expression and help factor it accurately. The quadratic formula is powerful in verifying possible roots when checker synthetic division results.

Polynomial Factoring

Factoring a polynomial involves rewriting it as a product of simpler polynomials. Combined with the Rational Root Theorem, synthetic division, and remaining root-solving, factoring completes the zero-finding mission of a polynomial. From our exercise:

• We found one root to begin - \(x = 1\). Using synthetic division, the quotient is \(4x^2 + 4x - 3\).
• By solving the quadratic using the quadratic formula, we found more roots: \(x = \frac{1}{2}\), and \(x = -\frac{3}{2}\).
• The final step is to express the polynomial's factored form as \(P(x) = 4(x - 1)(x - \frac{1}{2})(x + \frac{3}{2})\).
• Simplify by multiplying the constants together to refine it to \((x - 1)(2x - 1)(2x + 3)\).

This comprehensive approach ensures the polynomial matches its original form \(4x^3 - 7x + 3\) when expanded. Always verify by expanding to check the initial polynomial is accurately matched. Factoring reveals hidden structures, making polynomial equations easier to solve and understand.