Problem 72 - Elementary Algebra

Question

Use any of the factoring methods to factor. Identify any prime polynomials. $$ q^{2}+20 q+100 $$

Step-by-Step Solution

Answer

(q+10)^{2}

Step 1 - Identify the quadratic

The given polynomial is a quadratic expression of the form q^{2}+20q+100.

Step 2 - Find two numbers that multiply to the constant term and add to the coefficient of the linear term

To factor this quadratic expression, look for two numbers that multiply to the constant term 100 and add to the coefficient of the linear term 20. The numbers 10 and 10 satisfy both conditions: 10 * 10 = 100 and 10 + 10 = 20.

Step 3 - Write the factors

Using the numbers found, we can write the quadratic expression as (q+10)(q+10). This can be further simplified to (q+10)^{2}.

Step 4 - Identify if the polynomial is prime

Since we were able to factor the polynomial q^{2}+20q+100, it is not a prime polynomial.

Key Concepts

quadratic expression

A quadratic expression is a polynomial of degree two. It usually comes in the form
\[ ax^2 + bx + c \]
where

• a
• bx
• x^2
• c

are constants. In our example,
\[ q^2 + 20q + 100 \]
is the quadratic expression. The highest power of the variable

factoring methods

Factoring polynomials is the process of breaking down a polynomial into the product of simpler polynomials. Different methods exist to factorize polynomials. It depends on the specific type and form of the polynomial.

prime polynomials

A prime polynomial is one that cannot be factored into simpler polynomials over the integers. In other words, it is 'not factorable.' It is similar to prime numbers.

constant term

The constant term in a polynomial is the term with no variable attached to it. It is a standalone number.