Question

Factor each polynomial completely. If the polynomial cannot be factored, say it is prime. $$ 7\left(x^{2}-6 x+9\right)+5(x-3) $$

Short answer

The polynomial is prime and cannot be factored.

Step-by-step solution

- Distribute coefficients

First, distribute the coefficients inside the parentheses: distribute 7: 7(x^{2} - 6x + 9) = 7x^{2} - 42x + 63 distribute 5: 5(x - 3) = 5x - 15

- Combine like terms

Now, add the two resulting expressions together: 7x^2 - 42x + 63 + 5x - 15 Combine the like terms: 7x^2 - 37x + 48

- Factor polynomial

To factor, look for two numbers that multiply to (7 * 48) = 336 and add to -37. Factor the quadratic expression: (7x^2 - 37x + 48) Upon checking the factors, observe no such pair exists. Thus, the polynomial is prime.

Key concepts

distributing coefficients

To start factoring the polynomial, we need to distribute the coefficients to each term inside the parentheses. Distributing coefficients means multiplying every term inside the parentheses by the coefficient outside.
In our example, distribute the 7 to each term in \((x^{2} - 6x + 9)\):
$$ 7(x^{2} - 6x + 9) = 7x^{2} - 42x + 63 $$
Next, distribute the 5 to each term in \(x - 3\):
$$ 5(x - 3) = 5x - 15 $$
This step ensures we have eliminated all parentheses, simplifying further operations.

combining like terms

Once the coefficients are distributed, the next task is to combine like terms. Like terms have the same variable raised to the same power.
In our example, first write down all the terms together:
$$ 7x^{2} - 42x + 63 + 5x - 15 $$
Identify and combine the like terms separately:

• The \{x^2} terms: 7x^{2}
• The \{x} terms: -42x + 5x = -37x
• The constant terms: 63 - 15 = 48

Combining these, we get:
$$ 7x^{2} - 37x + 48 $$

prime polynomial

After combining like terms, we will attempt to factor the polynomial. Factoring involves finding two numbers that multiply to a certain constant and add up to the middle coefficient.
Our polynomial at this stage is:
$$ 7x^{2} - 37x + 48 $$
We need two numbers that multiply to \(7 * 48 = 336\) and add to -37. Despite attempting all factor pairs, no pair exists that satisfies both conditions. Thus, we conclude that the polynomial is prime.
A **prime polynomial** is one that cannot be factored over the set of integers. This means it does not break down further, making it similar to prime numbers in arithmetic.

quadratic expressions

Understanding **quadratic expressions** is crucial for factoring polynomials. A quadratic expression generally follows the form:
$$ ax^{2} + bx + c $$
where \{a} , \{b} , and \{c} are constants.
In our exercise, the quadratic expression we dealt with was:
$$ 7x^{2} - 37x + 48 $$
Key points when working with quadratic expressions:

• Identify coefficients \{a} , \{b} , and \{c}.
• Check if the expression can be factored.

If it can’t be factored, like in this case, it's considered a prime polynomial, signifying the simplest form of the quadratic expression. This is a fundamental aspect of understanding polynomial operations.