MATH 1201-01-College Algebra AY2026 T4 Unit 3 Discussion |
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| 100% correct
1. In 3-4 sentences, explain the Fundamental Theorem of Algebra with a suitable example.
Then find a polynomial with integer coefficients with roots Discuss the degree and roots of the
polynomial. Response to Q1:
The Fundamental Theorem of Algebra states that “every non-constant polynomial equation
with complex coefficients has at least one complex root. In other words, a polynomial of degree
n has exactly n roots, which may be real or complex, and some roots may repeat” (Abramson,
2021).
For example, the equation x^2 + 1 = 0 has no real roots, but it has two complex roots, i and -i.
This illustrates the Fundamental Theorem of Algebra, which guarantees the existence of
complex roots for all non-constant polynomial equations with complex coefficients.
To find a polynomial with integer coefficients with roots (sqrt {2}, 0); we can use the fact that
the product of the roots of a polynomial is equal to the constant term divided by the leading
coefficient. Therefore, if we have two roots, we can write the polynomial as (x - r1)(x - r2),
where r1 and r2 are the roots.
In this case, the polynomial with roots (sqrt {2}, 0) is (x - sqrt {2}) (x - 0) = x^2 - sqrt {2} x. This
polynomial has degree 2 and two roots, sqrt {2} and 0.
2. Peter wants to draw clear demarcation lines in a shopping mall’s car parking zone. The area of
the parking zone is (7x2-63) ft2. The area required to park one car is (x2-6x+9) ft2. Determine
the total number of cars that can be parked in the parking zone. What would happen if x ≤ 3?
Discuss the solution for some possible values of x.
{Hint: Simplify {(7x2-63)/(x2-6x+9)}}
Response to Q2:
To determine the total number of cars that can be parked in the parking zone, we need to divide
the total area of the parking zone by the area required to park one car. We can simplify the
expression first:
Verified study complete Solutions | A+ Graded | 2026 Updates
| 100% correct
1. In 3-4 sentences, explain the Fundamental Theorem of Algebra with a suitable example.
Then find a polynomial with integer coefficients with roots Discuss the degree and roots of the
polynomial. Response to Q1:
The Fundamental Theorem of Algebra states that “every non-constant polynomial equation
with complex coefficients has at least one complex root. In other words, a polynomial of degree
n has exactly n roots, which may be real or complex, and some roots may repeat” (Abramson,
2021).
For example, the equation x^2 + 1 = 0 has no real roots, but it has two complex roots, i and -i.
This illustrates the Fundamental Theorem of Algebra, which guarantees the existence of
complex roots for all non-constant polynomial equations with complex coefficients.
To find a polynomial with integer coefficients with roots (sqrt {2}, 0); we can use the fact that
the product of the roots of a polynomial is equal to the constant term divided by the leading
coefficient. Therefore, if we have two roots, we can write the polynomial as (x - r1)(x - r2),
where r1 and r2 are the roots.
In this case, the polynomial with roots (sqrt {2}, 0) is (x - sqrt {2}) (x - 0) = x^2 - sqrt {2} x. This
polynomial has degree 2 and two roots, sqrt {2} and 0.
2. Peter wants to draw clear demarcation lines in a shopping mall’s car parking zone. The area of
the parking zone is (7x2-63) ft2. The area required to park one car is (x2-6x+9) ft2. Determine
the total number of cars that can be parked in the parking zone. What would happen if x ≤ 3?
Discuss the solution for some possible values of x.
{Hint: Simplify {(7x2-63)/(x2-6x+9)}}
Response to Q2:
To determine the total number of cars that can be parked in the parking zone, we need to divide
the total area of the parking zone by the area required to park one car. We can simplify the
expression first:
