Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ 3\left(x^{2}+10 x+25\right)-4(x+5) $$

Short answer

Prime

Step-by-step solution

Expand Both Expressions

First, expand the polynomial. Start with the terms in the parentheses.For the first term: \[3(x^2 + 10x + 25) = 3x^2 + 30x + 75\] For the second term: \[-4(x + 5) = -4x - 20\]

Combine Like Terms

Combine the expanded terms: \[3x^2 + 30x + 75 - 4x - 20\]Combine like terms:\[3x^2 + (30x - 4x) + (75 - 20)\]This simplifies to:\[3x^2 + 26x + 55\]

Trial Factorization

Try to factor the resulting polynomial:\[3x^2 + 26x + 55\]To factorize this, look for two numbers that multiply to \(3 \times 55 = 165\) and add to 26. Since there are no such integer pairs, this polynomial cannot be factored further over the integers.

Key concepts

Polynomial Expansion

First, let's talk about polynomial expansion. This is the process where you remove the parentheses and distribute any coefficients to each term inside. Think of it like opening up a present, revealing all its components.
In our exercise, we started with the polynomial expression \[ 3(x^2 + 10x + 25) - 4(x + 5) \].
We need to distribute the 3 to each term inside the first parentheses and the -4 to each term inside the second parentheses. This yields:
\[ 3(x^2) + 3(10x) + 3(25) - 4(x) - 4(5) \] which simplifies to \[ 3x^2 + 30x + 75 - 4x - 20 \]

Combining Like Terms

After expanding, the next step is to combine like terms. Like terms are terms that have the same variables raised to the same power. They can be added or subtracted from each other.
For example, in the polynomial \[ 3x^2 + 30x + 75 - 4x - 20 \], we can combine:

• \[ 30x \] and \[ -4x \]
• \[ 75 \] and \[ -20 \]

This simplification results in: \[ 3x^2 + 26x + 55 \]. Combining like terms helps to simplify the polynomial, making it easier to handle in subsequent steps.

Prime Polynomial Identification

Now, let’s understand prime polynomial identification. A polynomial is considered 'prime' if it cannot be factored further over the integers. It's like a prime number in arithmetic, but applied to polynomials.
With our polynomial \[ 3x^2 + 26x + 55 \], we're trying to see if it can be broken down into simpler polynomials that multiply together to give the original one.
We check for integer pairs that multiply to the product of the leading coefficient (3) and the constant term (55), which is 165, and add up to the middle term coefficient, 26.
Since no such pairs exist, we can identify that \[ 3x^2 + 26x + 55 \] is prime. This means it cannot be factored further.

Trial Factorization

Trial factorization is the process of testing possible pairs of factors to see if they can break down the polynomial.
For our polynomial \[ 3x^2 + 26x + 55 \], we need two numbers that multiply to \[ 3 \times 55 = 165 \] and sum up to 26.
This involves checking various pairs like \[ (1, 165) \], \[ (3, 55) \], \[ (5, 33) \], and so on.
Since none of these pairs sum to 26, our trial factorization efforts show that the polynomial is prime.
Remember, trial factorization requires some guessing and checking, which can often affirm the polynomial's prime nature if no valid pairs are found.