h(x) =x + x - 9x - 9 2 h(x)=Dx 6.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Factoring the Polynomial**
For the polynomial below, \(-1\) is a zero.
\[ h(x) = x^3 + x^2 - 9x - 9 \]
**Objective:** Express \( h(x) \) as a product of linear factors.
**Explanation:**
To express \( h(x) \) in factored form, we need to use the fact that \(-1\) is a zero of the polynomial. This means that \((x + 1)\) is a factor of the polynomial. We can perform polynomial division or synthetic division to divide \( h(x) \) by \((x + 1)\) and find the remaining factors.
**Detailed Steps:**
1. **Synthetic or Polynomial Division:** Divide \( h(x) \) by \((x + 1)\) to find the quotient polynomial.
2. **Quotient Examination:** Once divided, the quotient can be further factored, possibly using quadratic formula or other factoring techniques if necessary.
3. **Express as Linear Factors:** Ultimately, express \( h(x) \) as a product of linear factors using the factor found from division and the zero.
This process helps in finding all the zeros of the polynomial and expressing it in a completely factored form, which is useful in solving, graphing, and understanding polynomial functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F29ce3a4c-cb07-4c4c-bb91-a57b2d1f76d3%2Fa66dffb8-2724-4099-9408-8b7c8d98684e%2Fsazjr9r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Factoring the Polynomial**
For the polynomial below, \(-1\) is a zero.
\[ h(x) = x^3 + x^2 - 9x - 9 \]
**Objective:** Express \( h(x) \) as a product of linear factors.
**Explanation:**
To express \( h(x) \) in factored form, we need to use the fact that \(-1\) is a zero of the polynomial. This means that \((x + 1)\) is a factor of the polynomial. We can perform polynomial division or synthetic division to divide \( h(x) \) by \((x + 1)\) and find the remaining factors.
**Detailed Steps:**
1. **Synthetic or Polynomial Division:** Divide \( h(x) \) by \((x + 1)\) to find the quotient polynomial.
2. **Quotient Examination:** Once divided, the quotient can be further factored, possibly using quadratic formula or other factoring techniques if necessary.
3. **Express as Linear Factors:** Ultimately, express \( h(x) \) as a product of linear factors using the factor found from division and the zero.
This process helps in finding all the zeros of the polynomial and expressing it in a completely factored form, which is useful in solving, graphing, and understanding polynomial functions.
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