Examples of prime factor in the following topics:
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- Such prime numbers are called prime factors.
- Therefore, 2 and 3 are prime factors of 6.
- However, 6 is not a prime factor.
- In this case, we must reduce 6 to its prime factors as well.
- We have now found factors for 12 that are all prime numbers.
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- For each of the denominators, we find all the prime factors—i.e., the prime numbers that multiply to give that number.
- If you are not familiar with the concept of prime factors, it may take a few minutes to get used to. 2⋅2⋅3 is 12 broken into its prime factors: that is, it is the list of prime numbers that when multiplied together yield 12.
- Similarly, the prime factors of 30 are 2, 3, and 5.
- Similarly, any number whose prime factors include a 2, a 3, and a 5 will be a multiple of 30.
- Finding the prime factors of the denominators of two fractions enables us to find a common denominator.
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- Factoring by grouping divides the terms in a polynomial into groups, which can be factored using the greatest common factor.
- Factor out the greatest common factor, 4x(x+5)+3y(x+5).
- The aim of factoring is to reduce objects to "basic building blocks", such as integers to prime numbers, or polynomials to irreducible polynomials.
- One way to factor polynomials is factoring by grouping.
- Both groups share the same factor (x+5), so the polynomial is factored as:
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- We can factor quadratic equations of the form ax2+bx+c by first finding the factors of the constant c.
- This leads to the factored form:
- First, we factor a, which has one pair of factors 3 and 2.
- Then we factor the constant c, which has one pair of factors 2 and 4.
- Using these factored sets, we assemble the final factored form of the quadratic
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- They describe musical intervals, appear in formulas counting prime numbers, inform some models in psychophysics, and can aid in forensic accounting.
- Natural logarithms are closely linked to counting prime numbers (2,3,5,7 ...), an important topic in number theory.
- The prime number theorem states that for large enough N, the probability that a random integer not greater than N is prime is very close to log(N)1.
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- When a trinomial is a perfect square, it can be factored into two equal binomials.
- It is important to be able to recognize such trinomials, so that they can the be factored as a perfect square.
- If you are attempting to to factor a trinomial and realize that it is a perfect square, the factoring becomes much easier to do.
- Since the middle term is twice 4⋅x, this must be a perfect square trinomial, and we can factor it as:
- Evaluate whether a quadratic equation is a perfect square and factor it accordingly if it is
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- When factoring, things are pulled apart.
- There are four basic types of factoring.
- The common factor is 3.
- This is the simplest kind of factoring.
- Therefore it factors as (x+5)(x−5).
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- To factor an expression means to rewrite it so that it is the product of factors.
- The reverse process is called factoring.
- Factoring is useful to help solve an equation of the form:
- Again, imagine you want to factor x2−7x+12.
- We attempt to factor the quadratic.
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- The factored form of a polynomial reveals its zeros, which are defined as points where the function touches the x-axis.
- The factored form of a polynomial can reveal where the function crosses the x-axis.
- In general, we know from the remainder theorem that a is a zero of f(x) if and only if x−a divides f(x). Thus if we can factor f(x) in polynomials of as small a degree as possible, we know its zeros by looking at all linear terms in the factorization.
- This is why factorization is so important: to be able to recognize the zeros of a polynomial quickly.
- Use the factored form of a polynomial to find its zeros