Problem 13 - Elementary Algebra

Question

Use the guess and check method to factor. Identify any prime polynomials. $$ x^{2}+14 x+49 $$

Step-by-Step Solution

Answer

(x + 7)(x + 7)

Step 1: Understand the Polynomial

The given polynomial is a quadratic equation in the form of dioceed of thentu polynomial and come up with potential factor pairs whose product equals 49 and sum equals 14.

Step 3: Multiply to Check

Multiply the factor pairs to verify their product equals the original factors.

Step 4: Solve for Coefficients

Since Step is checking and confirming that all the criteria match.

Key Concepts

guess and check method

The 'guess and check method' is a popular way to factor quadratic equations. It's very intuitive and straightforward. The process involves making educated guesses about the factors of a quadratic polynomial and verifying them by checking the product. Start with a quadratic equation in the standard form:
Ax^2 + Bx + C.
The goal here is to express this polynomial as the product of two binomials:
(Dx + E)(Fx + G).
Here’s a step-by-step breakdown: List the factors of C.
- Guess pairs of factors.
- Substitute these pairs into the binomials.
- Expand the binomials to see if you retrieve the original quadratic equation.
For example, for the quadratic equation [ x^2 + 14x + 49 ]
- List factors of 49: [ 1, 49 ] and [ 7, 7 ]
- Guess pairs of factors until you find the correct binomials: (x + 7)(x + 7)
- Expand to verify: (x + 7)(x + 7) = x^2 + 14x + 49
The guess and check method ensures that each guess is verified, making it a reliable method for factoring quadratic equations.

prime polynomial

A prime polynomial is a polynomial that cannot be factored over the set of integers. In terms of quadratic equations, this means the quadratic polynomial cannot be written as a product of two simpler polynomials with integer coefficients. Identifying a prime polynomial is essential because it tells you when you can stop trying to factorize it.
Consider the polynomial: [ x^2 + 14x + 49 ]
If following the guess and check method doesn't result in valid integer pairs, then the polynomial is prime. However, our example [ x^2 + 14x + 49 ] is not prime because it can be factored into [ (x + 7)(x + 7) ]. When you are certain a polynomial can’t be factored using any integers, you can safely conclude it is a prime polynomial.

quadratic equation

A quadratic equation is a second-degree polynomial, meaning the highest exponent of the variable is 2. Generally, it is written in the form of [ Ax^2 + Bx + C = 0 ] where A, B, and C are constants. Quadratic equations are fundamental in algebra and appear in various real-world scenarios, such as physics, engineering, and economics.
Key features:
- The graph of a quadratic equation is a parabola.
- A quadratic equation can have 0, 1, or 2 real roots.
Ways to solve a quadratic equation include:

• Factoring (like in our example)
• The quadratic formula
• Completing the square
  The given quadratic equation [ x^2 + 14x + 49 ] is a perfect square trinomial. When we factor this quadratic, we find: [ (x + 7)(x + 7) ] = [ (x + 7)^2 ]. Understanding quadratic equations is essential to mastering algebra.