Problem 8 - Elementary Algebra

Question

Use a pattern to factor. Check. Identify any prime polynomials. $$ c^{2}-14 c+49 $$

Step-by-Step Solution

Answer

\[ (c - 7)^2 \]

Step 1 - Identify the quadratic form

Notice that the given polynomial is in the standard quadratic form: \[ c^2 - 14c + 49 \]

Step 2 - Recognize the perfect square trinomial

To factor the polynomial, recognize it as a perfect square trinomial. A perfect square trinomial has the form \[ (a - b)^2 = a^2 - 2ab + b^2 \]. In this case, compare it with our expression \[ c^2 - 14c + 49 \].

Step 3 - Identify 'a' and 'b'

Here, \[ a = c \] and \[ b = 7 \], so the polynomial can be written as: \[ c^2 - 2(7)(c) + 7^2 \]

Step 4 - Factor the polynomial

Rewrite the polynomial as: \[ (c - 7)^2 \]

Step 5 - Check the factorization

Expand \[ (c - 7)^2 \] to verify it matches the original polynomial: \[ (c - 7)(c - 7) = c^2 - 7c - 7c + 49 = c^2 - 14c + 49 \]

Step 6 - Identify prime polynomials

Since we were able to factor the polynomial, it is not prime. Therefore, \[ c^2 - 14 c + 49 \] is not a prime polynomial.

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a specific type of polynomial. It's created by squaring a binomial. This means it takes the form \((a - b)^2 = a^2 - 2ab + b^2\).

In our example, \(c^2 - 14c + 49\), we need to see if it fits this form.

Comparing it, we notice that:

• \(a^2 = c^2\) means \(a = c\)
• \(-2ab = -14c\) leads us to find \(b = 7\)
• \(b^2 = 49\)

So, we can recognize that \(c^2 - 14c + 49\) is indeed a perfect square trinomial. This helps us greatly in its factorization process.

Quadratic Form

The quadratic form is a type of polynomial with three parts: \(ax^2 + bx + c\). It's called 'quadratic' because the highest degree of the variable (in this case, c) is squared.

In our example \(c^2 - 14c + 49\), it matches the standard quadratic form where:

• \(a = 1\)
• \(b = -14\)
• \(c = 49\)

Identifying this structure helps us apply different techniques like completing the square or using special factorization patterns.

Factorization Process

Factoring polynomials involves breaking them down into simpler pieces called 'factors'. If we recognize a pattern like a perfect square trinomial, factorization becomes easier.

Follow these basic steps to factor a polynomial like \(c^2 - 14c + 49\):

1. Identify the quadratic form.
2. Recognize the perfect square trinomial by comparing it with \( (a - b)^2 \).
3. Determine values for \(a\) and \(b\).
4. Rewrite the polynomial as \((c - 7)^2 \).

After these steps, always check by expanding back to ensure it matches the original polynomial. This confirms our factorization.

Prime Polynomials

A prime polynomial cannot be factored into simpler polynomials with integer coefficients.

To determine if a polynomial is prime:

• Attempt to factor it.
• If it can be factored (like our example \(c^2 - 14c + 49 = (c - 7)^2\)), it is not prime.
• If no factorization is possible, it is a prime polynomial.

Our example—having a clear factorization—shows \(c^2 - 14c + 49\) is not a prime polynomial.