Problem 43 - Algebra and Trigonometry

Question

Find all rational zeros of the polynomial, and write the polynomial in factored form. $$ P(x)=x^{5}+3 x^{4}-9 x^{3}-31 x^{2}+36 $$

Step-by-Step Solution

Answer

The polynomial has rational zeros at \( x = 1, x = 2 \), and the factored form is \( (x-1)(x+1)(x^2+6x+18) \).

Step 1: Identify Possible Rational Zeros

To find the rational zeros of the polynomial, use the Rational Root Theorem. This theorem states that any rational zero, \( p/q \), of the polynomial \( P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 \), where \( a_0 \) is the constant term and \( a_n \) is the leading coefficient, must be such that \( p \) divides \( a_0 \) and \( q \) divides \( a_n \). Here, \( a_0 = 36 \) and \( a_n = 1 \), so the possible rational zeros are the factors of 36: \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 9, \pm 12, \pm 18, \pm 36 \).

Step 2: Test Possible Rational Zeros

Substitute each possible rational zero into the polynomial \( P(x) \) to determine which ones are actual zeros. Use synthetic division or direct substitution to see which values of \( x \) result in \( P(x) = 0 \). Start with smaller numbers for efficiency. 1. \( P(1) = 1^5 + 3(1)^4 - 9(1)^3 - 31(1)^2 + 36 = 0 \) 2. \( P(-1) = (-1)^5 + 3(-1)^4 - 9(-1)^3 - 31(-1)^2 + 36 eq 0 \)3. Continue testing until all possibilities are examined.

Step 3: Factor the Polynomial

Once a zero is found, use it to factor the polynomial. Since \( x = 1 \) is a zero, \( x - 1 \) is a factor. Use synthetic division to divide \( P(x) \) by \( x - 1 \) to find the quotient. After synthetic division: \( P(x) = (x - 1)(x^4 + 4x^3 - 5x^2 - 36x) + 0 \)Continue factoring \( x^4 + 4x^3 - 5x^2 - 36x \) by finding additional zeros and using synthetic division until it is fully factored.

Step 4: Find and Verify Additional Zeros

Repeat the process of finding zeros for the remaining polynomial \( x^4 + 4x^3 - 5x^2 - 36x \). 1. Add \( x^2 \) from both sides: \( x(x^3 + 4x^2 - 5x - 36) \)2. Test potential rational zeros in the cubic \( x^3 + 4x^2 - 5x - 36 \). 3. Use synthetic division to confirm factors, such as finding \( x = 2 \) for \( x^3 + 4x^2 - 5x - 36 = 0 \).4. After division: \( x^4 + 4x^3 - 5x^2 - 36x = x(x - 1)(x^2 + 6x + 18) \).

Step 5: Combine and Simplify to Final Factored Form

Once all rational and verified real zeros are determined, write the polynomial in fully factored form.\( P(x) = x(x + 1)(x - 2)(x^2 + 6x + 18) \).Verify the factorization by expanding the factored form and ensuring it matches the original polynomial.

Key Concepts

Polynomial Factoring

Polynomial factoring is an important algebraic process that breaks down a polynomial into simpler elements called factors. Think of it like breaking a large Lego build into smaller pieces. When you factor a polynomial, you are expressing it as a product of other polynomials. This is handy for solving equations, simplifying expressions, and finding zeros or roots of the polynomial.

To factor a polynomial, you'll often be searching for "roots," which are values of the variable that make the polynomial equal zero. For instance, in our polynomial, we want to find these roots and express the polynomial in terms of its factors.

Synthetic Division

Synthetic division is a simplified form of polynomial division, similar to long division but more efficient and quicker. It comes in handy when testing potential rational zeros or simplifying polynomials during the factoring process. Let's say we find a zero, like 1, in our example exercise. Once we identify this zero, synthetic division helps us divide the original polynomial by the corresponding linear factor (i.e., \(x - 1\)).

• Write down the coefficients of the polynomial. For \(x^5 + 3x^4 - 9x^3 - 31x^2 + 36\), they are 1, 3, -9, -31, and 36.
• Place the zero found (1 in this case) to the left and perform operations to bring a downscaled polynomial as a result.
• This allows us to find the next factor which simplifies the original polynomial step by step.

The outcome is a reduced polynomial that we continue factoring until it is fully simplified.

Rational Zeros

Rational zeros are the values of \(x\) where the polynomial equals zero, and they are rational numbers that can be expressed as fractions. The Rational Root Theorem assists in identifying these potential zeros. It states that for a given polynomial, any rational zero, \(\frac{p}{q}\), must have \(p\) as a factor of the constant term and \(q\) as a factor of the leading coefficient.For example, in our exercise, the constant term is 36, and the leading coefficient is 1. Possible rational zeros for our polynomial would be the factors of 36: \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 9, \pm 12, \pm 18, \pm 36 \).

• Test each potential candidate in the original polynomial.
• Use synthetic division or direct substitution to check which candidate makes the polynomial zero.

Discovering these rational zeros is essential in breaking down the polynomial further into factors.

Factored Form of Polynomials

The factored form of a polynomial is an expression of the polynomial as a product of its factors. This is immensely useful for understanding the roots of the polynomial and for various algebraic manipulations.In the given exercise, by finding the rational zeros and using synthetic division, we reduced and factored the polynomial step by step. Eventually, we reached the factored form \[ P(x) = x(x + 1)(x - 2)(x^2 + 6x + 18) \].Each term in this expression corresponds to a zero (or root) of the original polynomial:

• \(x = 0\)
• \(x + 1 = 0 \Rightarrow x = -1\)
• \(x - 2 = 0 \Rightarrow x = 2\)

Expressing the polynomial in factored form verifies that you have found all possible zeros, providing insights into the behavior of the polynomial's graph on a coordinate plane.