Problem 18 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

(c - 7d)^2. Verified by expanding the binomial and confirming no prime polynomials remain.

Step 1 - Recognize the Square Pattern

Notice that the given polynomial is in the form of a perfect square trinomial, which can be written as c^2 - 2ab + b^2. Compare this with c^2 - 14cd + 49d^2.

Step 2 - Break down the middle term

Here, you can observe that - 14cd actually represents the middle term . In the perfect square trinomial pattern - 2ab, the -14cd corresponds to -2ab. This implies -2ab = -14cd.

Step 3 - Identify 'a' and 'b'

From Step 2, we can identify 'a' and 'b' such that a = c and b =7d. Hence, the expression can be written in the form a^2 + 2ab + b^2 again.

Step 4 - Write as Perfect Square Binomial

Hence, recognizing the perfect square binomial, the expression can be factored. Factor the expression c^2 - 14cd + 49d^2 as (c -7d)^2.

Step 5 - Check by Expanding

To verify, expand the binomial (c -7d)(c -7d) and it results back to the original polynomial Check: (c -7d)(c -7d) = c^2 - 7cd -7cd + 49d^2 =c^2 - 14cd + 49d^2.

Step 6 - Identify Prime Polynomials

After factoring, (c -7d)^2 cannot be factored further thus is considered prime. Conclusion: The given polynomial was factors correctly without leaving any prime polynomial

Key Concepts

Perfect Square Trinomial

Let's start by exploring what a perfect square trinomial is and how to recognize it. In algebra, a perfect square trinomial is a special type of polynomial that takes the form \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\). These can be factored into \((a + b)^2\) or \((a - b)^2\) respectively. In our example, the given polynomial is \(c^2 - 14cd + 49d^2\). Recognizing the pattern, we see it fits the form \(a^2 - 2ab + b^2\).

Here’s a step-by-step process to identify and factor it:

• Identify if the polynomial is a trinomial (three terms).
• Check if it fits the perfect square trinomial pattern.
• Compare each component with the forms \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\).

In our specific case, we identify \(a = c\) and \(b = 7d\). Therefore, the polynomial \(c^2 - 14cd + 49d^2\) can be factored into \((c - 7d)^2\).

To check if the factorization is correct, expand \((c - 7d)^2\). You get back the original trinomial, confirming your factorization.

Prime Polynomial

A prime polynomial is a polynomial that cannot be factored further into polynomials of lower degrees. Think of it like a prime number in arithmetic which can't be divided by any numbers other than 1 and itself. After factoring a polynomial, we always need to check whether the resulting factors can be broken down further.

In the example given, after factoring \(c^2 - 14cd + 49d^2\) into \((c - 7d)^2\), you see it's now in its simplest form. Factors like \((x + y)\) or \((x - y)\) where further factorization isn't possible indicate the polynomial is prime.

• Factorize the polynomial into binomials.
• Check each resulting binomial to see if it can be factored further.
• If no further factorization is possible, then it's prime.

Hence, once you factor \(c^2 - 14cd + 49d^2\) to \((c - 7d)^2\), it cannot be factored anymore, confirming it's a prime polynomial.

Factoring Binomials

Factoring binomials is one of the fundamental skills in algebra. It involves expressing a binomial (a polynomial with two terms) as a product of two factors. The binomials can often be further simplified to forms such as \((x + y)\) or \((x - y)\). There's a variety of methods for factoring binomials depending on the specific forms they take.

Here are some common techniques:

• Difference of Squares: For expressions like \(a^2 - b^2\), factor them as \((a + b)(a - b)\).
• Sum and Difference of Cubes: For \(a^3 + b^3\) and \(a^3 - b^3\), factor them as \((a + b)(a^2 - ab + b^2)\) and \((a - b)(a^2 + ab + b^2)\) respectively.
• Grouping: When you have four terms, you can sometimes group them into pairs and factor by grouping.

In our original example, after recognizing \(c^2 - 14cd + 49d^2\) as a perfect square trinomial, we factored it directly as \((c - 7d)^2\). Once you get comfortable with these patterns, factoring binomials becomes straightforward and even enjoyable!