Factorization Practice Test Quiz

Explanation
The number 10 is not a factor of 15 because a factor of a number is a whole number that can be divided evenly into that number. When we divide 15 by 10, we get a remainder of 5, which means 10 is not a factor of 15.

Explanation
The number 100 is not a multiple of 15 because it cannot be divided evenly by 15. A multiple of a number is obtained by multiplying that number by any whole number. In this case, 15 multiplied by 1 gives 15, 15 multiplied by 2 gives 30, and so on. However, when we multiply 15 by any whole number, we will never get 100 as the result. Therefore, 100 is not a multiple of 15.

Explanation
Completing the square is a method for factorizing quadratic equations. This involves manipulating the equation to create a perfect square trinomial, which can then be factored easily. By adding or subtracting a constant term to both sides of the equation, the quadratic expression can be expressed as the square of a binomial, allowing for easy factorization. This method is commonly used when solving quadratic equations or simplifying expressions involving quadratic terms.

Explanation
The quadratic formula is used to factorize quadratic equations. It is derived from completing the square method and can be used to find the roots or solutions of a quadratic equation. The formula is x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation. By substituting the values of a, b, and c into the formula and simplifying, the roots of the equation can be determined.

Explanation
Divisor domain is the correct answer because integral domains that share a similar property after factorization are called divisor domains. In a divisor domain, every nonzero element can be factored into irreducible elements, and this factorization is unique up to associates. This property allows for the cancellation of common factors in a divisor domain, making it a useful concept in algebraic number theory and commutative algebra.

Explanation
The number 15 is not a prime number because it is divisible by both 3 and 5. A prime number is a number that is only divisible by 1 and itself, but 15 can be divided by other numbers as well. Therefore, 15 is not a prime factor of 45.

Explanation
An integral domain is a commutative ring in which the product of any two non-zero elements is non-zero. When a domain is factorized, it means that it can be expressed as a product of irreducible elements. In an integral domain, every element can be factored in a unique way into irreducible elements, similar to prime factorization in the integers. Therefore, an integral domain is euclidean when factorized.

Explanation
The factors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, and 90. Out of these, the factors that are less than 10 are 1, 2, 3, 5, 6, and 9. Therefore, there are 6 factors of 90 that are less than 10.

Explanation
There are 3 prime factors of 20. The prime factors of 20 are 2, 2, and 5.

Explanation
The highest factor of 8 is 4 because a factor is a number that divides evenly into another number. In this case, 4 is the largest number that can divide evenly into 8 without leaving a remainder.