Problem 36 - Precalculus Mathematics for Calculus

Question

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=8 x^{3}+10 x^{2}-x-3$$

Step-by-Step Solution

Answer

The polynomial in factored form is \((x + 1)(2x - 1)(4x + 3)\).

Step 1: Identify possible rational roots using the Rational Root Theorem

The Rational Root Theorem states that any rational root of the polynomial \(P(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_0\) is of the form \(\frac{p}{q}\), where \(p\) is a factor of \(a_0\) (the constant term) and \(q\) is a factor of \(a_n\) (the leading coefficient). For \(P(x) = 8x^3 + 10x^2 - x - 3\), the possible values of \(p\) are the factors of \(-3\): \(\pm 1, \pm 3\). The possible values of \(q\) are the factors of \(8\): \(\pm 1, \pm 2, \pm 4, \pm 8\). Therefore, the possible rational roots are \(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{1}{8}, \pm 3, \pm \frac{3}{2}, \pm \frac{3}{4}, \pm \frac{3}{8}\).

Step 2: Test possible rational roots by substitution

To find which, if any, of these potential roots are actual roots, substitute them into \(P(x)\) and check if it equals zero. Start by testing simple integers like \(x = 1\) and \(x = -1\).- For \(x = 1: P(1) = 8(1)^3 + 10(1)^2 - 1 - 3 = 8 + 10 - 1 - 3 = 14\). Not zero.- For \(x = -1: P(-1) = 8(-1)^3 + 10(-1)^2 + 1 - 3 = -8 + 10 + 1 - 3 = 0\). \(x = -1\) is a root.

Step 3: Perform polynomial division to factor the polynomial

Now that we have a root \(x = -1\), perform synthetic or polynomial division to divide \(P(x)\) by \(x + 1\).Using synthetic division with \(-1\):\[\begin{array}{r|rrrr}-1 & 8 & 10 & -1 & -3 \ & & -8 & -2 & 3 \\hline & 8 & 2 & -3 & 0 \\end{array}\]This gives: \(8x^2 + 2x - 3\). The remainder is zero, indicating \(x + 1\) is a factor.

Step 4: Factor the quadratic further

Now, factor \(8x^2 + 2x - 3\). Find two numbers that multiply to \(8 \times -3 = -24\) and add to \(2\). These numbers are \(6\) and \(-4\). Rewrite the quadratic:\(8x^2 + 6x - 4x - 3\).Group and factor by grouping:\((8x^2 + 6x) + (-4x - 3)\).Factor each group:\(2x(4x + 3) - 1(4x + 3)\).Factor out the common factor \((4x + 3)\):\((2x - 1)(4x + 3)\).

Step 5: Write the polynomial in factored form

Now combine this result with the factor from Step 3:\(P(x) = (x + 1)(2x - 1)(4x + 3)\).

Key Concepts

Polynomial Division

Polynomial division is a technique used to divide one polynomial by another, similar to long division with numbers. It allows us to write a given polynomial as the product of another polynomial and a divisor, plus a remainder.
For example, when dealing with the polynomial \(P(x) = 8x^3 + 10x^2 - x - 3\) and aiming to divide it by \(x + 1\), you perform a division operation that helps to simplify or factor the polynomial.

• First, you determine how many times the first term of the divisor goes into the first term of the dividend.
• Then, multiply the entire divisor by that quotient and subtract it from the dividend.
• Next, you bring down the next term from the original dividend and repeat the process until all terms are exhausted.

The remainder of this division is essential. If it's zero, it means the divisor is indeed a factor of the original polynomial. In polynomial division, successfully dividing with a remainder of zero allows us to conclude that \(x+1\) is a divisor, simplifying our original polynomial.

Factored Form

Factored form of a polynomial means expressing it as a product of its factors. Finding the factored form is often necessary for solving polynomial equations efficiently.
In our example, after using synthetic division and factoring, we found that the factored form of \(P(x)\) is \((x + 1)(2x - 1)(4x + 3)\).

• To achieve this, first identify a root (for example, \(x = -1\)). This gives us one factor as \(x + 1\).
• Afterward, the remaining quadratic \(8x^2 + 2x - 3\) is factored further into \((2x - 1)(4x + 3)\).

This systematic breakdown reveals all the roots of the polynomial and allows easier solving of related equations. Understanding the factored form is crucial in algebra as it aids in both theoretical and practical applications, such as finding zeros or solving inequalities.

Synthetic Division

Synthetic division is a simplified method of dividing a polynomial by a linear binomial of the form \(x - c\) and is particularly helpful when identifying roots or simplifying polynomials.
Using synthetic division with the polynomial \(P(x) = 8x^3 + 10x^2 - x - 3\) and root \(x=-1\) proceeds as follows:

• Write down the coefficients of \(P(x)\) in a row: \(8, 10, -1, -3\).
• Place \(-1\) outside (the root you're testing).
• Bring down the leading coefficient (8) to start your answer line.
• Multiply this number by \(-1\) and add it to the next coefficient. Place the result beneath the line.
• Repeat this for all coefficients.

If the last number is a zero, \(x+1\) is a factor of \(P(x)\). This confirms the factor and helps continue factoring the remainder. Synthetic division is a quick and easy tool that provides insight into polynomial behavior and its roots efficiently.