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Magazine Chapter 23: Problem 22
Find all zeros of \(6 x^{4}+17 x^{3}+7 x^{2}+x-10 \operatorname{in} Q\). (This is a tedious high school algebra problem. You might use a bit of analytic geometry and calculus and make a graph, or use Newton's method to see which are the best candidates for zeros.)
Short Answer
Expert verified
The zeros of the polynomial are \(-2, \frac{1}{3}, 1, \text{and} -\frac{5}{2}\).
Step by step solution
01
Identify Potential Rational Zeros
Use the Rational Root Theorem, which states that any rational root of a polynomial equation with integer coefficients is of the form \( \frac{p}{q} \), where \( p \) is a factor of the constant term (-10) and \( q \) is a factor of the leading coefficient (6). So, potential rational zeros are: \( \pm 1, \pm 2, \pm 5, \pm 10, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}, \pm \frac{5}{2}, \pm \frac{5}{3}, \pm \frac{5}{6}, \pm \frac{10}{3}, \pm \frac{10}{6} \).
02
Test Potential Zeros Using Synthetic Division
Test each potential zero from Step 1. Start with \( x = 1 \) using synthetic division. Compute and check if the remainder equals zero. If not, move to the next potential zero.
03
Identify First Zero
Using synthetic division, find that \( x = -2 \) is a zero of the polynomial, showing that \( 6(-2)^4 + 17(-2)^3 + 7(-2)^2 + (-2) - 10 = 0 \). This gives us \( (x + 2) \) as a factor, and leaves us with \( 6x^3 + 5x^2 - 3x - 5 \).
04
Repeat Synthetic Division
Perform synthetic division again on the reduced polynomial \( 6x^3 + 5x^2 - 3x - 5 \) using the remaining candidates. Continue until another zero is found.
05
Identify Second Zero
Find that \( x = \frac{1}{3} \) is a zero. Perform synthetic division to confirm this zero, leading to a further reduced polynomial \( 6x^2 + 3x - 15 \).
06
Factor the Remaining Quadratic Polynomial
Factor or use the quadratic formula to solve \( 6x^2 + 3x - 15 = 0 \). This can be factored to \( (3x - 3)(2x + 5) = 0 \), giving us zeros: \( x = 1 \) and \( x = -\frac{5}{2} \).
07
Compile All Zeros
List all zeros found: \( x = -2, \frac{1}{3}, 1, \) and \( -\frac{5}{2} \). These are the zeros of the polynomial equation.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Synthetic Division
Synthetic division is a simplified form of dividing polynomials, particularly useful when dealing with a linear divisor. It is a quick method to evaluate the potential zeros of a polynomial, especially when using the Rational Root Theorem. Unlike long division, synthetic division requires less writing and fewer steps.
Here's how to perform synthetic division:
Here's how to perform synthetic division:
- Set up by writing down the coefficients of the polynomial you're dividing.
- Choose a candidate zero, a value you suspect could be a root (usually from those identified using the Rational Root Theorem).
- Begin the process by bringing the first coefficient straight down.
- Multiply this number by the candidate zero and add it to the next coefficient, repeating the process across all coefficients.
- If the last number (the remainder) is zero, your candidate zero is indeed a root of the polynomial.
Polynomial Zeros
Polynomial zeros, sometimes called roots or solutions, are the values of x that make the polynomial equal to zero. Finding these zeros is crucial because they are where the graph of the polynomial touches or crosses the x-axis.
There are several ways to find polynomial zeros:
There are several ways to find polynomial zeros:
- Applying the Rational Root Theorem gives potential rational zeros as fractional values derived from the factors of the polynomial's constant term and leading coefficient.
- Using synthetic division helps in verifying these potential zeros by checking if they result in zero remainders.
- For higher degree polynomials, a combination of guessing, checking, and analytical methods like factoring or using the quadratic formula is often employed.
Factoring Polynomials
Factoring polynomials involves expressing a polynomial as a product of simpler polynomials or factors, which can simplify the process of solving polynomial equations.
In the step-by-step solution provided, several factoring techniques appear:
In the step-by-step solution provided, several factoring techniques appear:
- Identifying linear factors using found zeros, such as \((x + 2)\), derived from the zero \(x = -2\).
- Using the reduced polynomial obtained from synthetic division for further factorization, like simplifying \(6x^3 + 5x^2 - 3x - 5\) into smaller components.
- Applying methods like the quadratic formula to factor the quadratic part, \(6x^2 + 3x - 15\), when simple factoring by inspection is not evident. This formula gives roots that can simplify into rational numbers, offering clear polynomial factors.
