Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 6: Problem 26
Find all of the zeros of each function. \(h(x)=10 x^{3}-17 x^{2}-7 x+2\)
Short Answer
Expert verified
The zeros of the function are \(-\frac{1}{2}\), \(\frac{1}{5}\), and \(2\).
Step by step solution
01
Identify Possible Rational Zeros
According to the Rational Root Theorem, the possible rational zeros of a polynomial are the factors of the constant term divided by the factors of the leading coefficient. For the function \(h(x)=10x^3 - 17x^2 - 7x + 2\), the constant term is 2 and the leading coefficient is 10. The factors of 2 are \(\pm 1, \pm 2\), and the factors of 10 are \(\pm 1, \pm 2, \pm 5, \pm 10\). The possible rational zeros are \(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{5}, \pm \frac{1}{10}, \pm 2\).
02
Use Synthetic Division
Test each possible rational zero using synthetic division. First, use \(+1\) as a test zero. Set up synthetic division for \(10x^3 - 17x^2 - 7x + 2\):\[\begin{array}{r|rrrr}1 & 10 & -17 & -7 & 2 \ & & 10 & -7 & -14 \\hline & 10 & -7 & -14 & -12\end{array}\]The remainder is not zero, so \(+1\) is not a zero. Repeat this process for each possible zero.
03
Find a Zero Using Synthetic Division
After testing, \(-\frac{1}{2}\) is found to be a zero using synthetic division:\[\begin{array}{r|rrrr}-\frac{1}{2} & 10 & -17 & -7 & 2 \ & & -5 & 11 & -2 \\hline & 10 & -22 & 4 & 0\end{array}\]The remainder is zero, so \(-\frac{1}{2}\) is a zero.
04
Factor the Polynomial
Use the zero found to factor \(h(x)\). We have \(h(x) = (x + \frac{1}{2})(10x^2 - 22x + 4)\). Simplify the quadratic by factoring out the greatest common divisor:\(10x^2 - 22x + 4 = 2(5x^2 - 11x + 2)\)Thus, \(h(x) = (x + \frac{1}{2})(2)(5x^2 - 11x + 2)\).
05
Factor the Quadratic Equation
Now, solve \(5x^2 - 11x + 2 = 0\) by factoring further. The quadratic factors as:\((5x - 1)(x - 2) = 0\)The solutions to this equation are \(x = \frac{1}{5}\) and \(x = 2\).
06
List All Zeros
The zeros of the original polynomial \(h(x)=10x^3 - 17x^2 - 7x + 2\) are: \(-\frac{1}{2}\), \(\frac{1}{5}\), and \(2\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Rational Root Theorem
The Rational Root Theorem is a valuable tool when finding the zeros of a polynomial function. In this case, for the polynomial \( h(x) = 10x^3 - 17x^2 - 7x + 2 \), the theorem helps us identify potential rational zeros. To apply the Rational Root Theorem, one needs to consider two things:
- The factors of the constant term of the polynomial, which is 2 in this example. The factors of 2 are \( \pm 1 \) and \( \pm 2 \).
- The factors of the leading coefficient, which is 10. The factors of 10 are \( \pm 1, \pm 2, \pm 5, \pm 10 \).
Synthetic Division
Synthetic division offers a simplified way to divide a polynomial by a linear factor of the form \( x - c \). When searching for zeros of \( h(x) = 10x^3 - 17x^2 - 7x + 2 \), one tries each potential zero identified by the Rational Root Theorem to see if it results in a remainder of zero, thereby confirming it as an actual zero.
To perform synthetic division:
To perform synthetic division:
- Write down the coefficients of the polynomial: 10, -17, -7, and 2.
- Select a potential zero, for example, -\(\frac{1}{2} \), and place it at the start.
- Carry out the synthetic division process by bringing down the first coefficient, multiplying and adding as per the structure of synthetic division.
- If the remainder is zero, \( c \) is a root. Otherwise, it is not.
Factorization
Factorization involves expressing a polynomial as a product of its factors, and it becomes easier once you have identified at least one zero using synthetic division. After identifying \( -\frac{1}{2} \) as a zero, \( h(x) \) could be factored into \( (x + \frac{1}{2})(10x^2 - 22x + 4) \).
- The factor \( x + \frac{1}{2} \) comes from the zero \( -\frac{1}{2} \). When a zero is found, \( (x - c) \) becomes a factor, here adjusted for the negative zero.
- The remaining polynomial, \( 10x^2 - 22x + 4 \), can often be further factored. Simplify by factoring out any greatest common divisor, leading to \( 2(5x^2 - 11x + 2) \).
- This step reduces the polynomial, making it easier to solve or further analyze.
Quadratic Equation
Once the polynomial is reduced to a quadratic equation, solving it can provide any additional zeros. The remaining quadratic \( 5x^2 - 11x + 2 \) was solved through factorization. This process involved:
- Looking for two numbers that multiply to the product of 5 (leading coefficient) and 2 (constant term), in this case, 10, and adding to -11 (middle term).
- Finding that these numbers are -1 and -10, suggesting the factorization \( (5x - 1)(x - 2) = 0 \).
- Each factor provides a solution or zero: solving \( 5x - 1 = 0 \) yields \( x = \frac{1}{5} \), and solving \( x - 2 = 0 \) yields \( x = 2 \).
