Problem 20 - Elementary Algebra

Question

Use a pattern to factor. Check. Identify any prime polynomials. $$ c^{18}-14 c^{9} d^{9}+49 d^{18} $$

Step-by-Step Solution

Answer

\[ c^{18} - 14 c^9 d^9 + 49 d^{18} = (c^9 - 7 d^9)^2 \] and \( c^9 - 7 d^9 \) is prime.

Step 1: Identify the structure of the polynomial

Recognize that the given polynomial is in the form of a quadratic pattern in terms of a variable. Notice that we can rewrite it in terms of \( c^9 \) and \( d^9 \): \( (c^9)^2 - 2 \times 7 \times c^9 \times d^9 + (7 d^9)^2 \).

Step 2: Rewrite using the quadratic form

Rewrite the polynomial as \[ (c^9)^2 - 14 c^9 d^9 + (7 d^9)^2 \].

Step 3: Factor using the perfect square trinomial

Identify that this is a perfect square trinomial: \[ (c^9 - 7 d^9)^2 \].

Step 4: Simplify the expression

Write the factored form of the polynomial as \[ (c^9 - 7 d^9)^2 \].

Step 5: Check for prime polynomials

Check the factor \( c^9 - 7 d^9 \). Notice that this polynomial cannot be factored further using integers, making it a prime polynomial.

Key Concepts

quadratic pattern

The term 'quadratic pattern' refers to recognizing when a polynomial fits the standard form of a quadratic equation, which is usually written as \[ax^2 + bx + c = 0\]. In many instances, polynomials can be rewritten to showcase this pattern, even if the variable appears in a different form. For the given polynomial \[c^{18} - 14c^9d^9 + 49d^{18}\], notice it doesn't fit the conventional quadratic instantly. However, by defining a new variable \[u = c^9\] and \[v = d^9\], we get \[u^2 - 2 \times 7 \times u \times v + (7v)^2\]. This transformation reveals the 'quadratic pattern' hidden inside our initial polynomial.

perfect square trinomial

A perfect square trinomial is a polynomial that can be written as the square of a binomial. In general, it takes the form \[a^2 + 2ab + b^2 = (a + b)^2\] or \[a^2 - 2ab + b^2 = (a - b)^2\]. Recognizing this pattern allows us to simplify and factor polynomials efficiently.
For the polynomial \[c^{18} - 14c^9d^9 + 49d^{18}\], notice how we can rewrite it as \[ (c^9)^2 - 2 \times 7 \times c^9 \times d^9 + (7d^9)^2\]. This fits perfectly into the form \[a^2 - 2ab + b^2 = (a - b)^2\]. Therefore, we identify that it's a perfect square trinomial and can be factored as \[(c^9 - 7d^9)^2\].

prime polynomial

A prime polynomial is one that cannot be factored further using integers. After factoring the polynomial \[c^{18} - 14 c^9 d^9 + 49 d^{18}\] into \[(c^9 - 7 d^9)^2\], we need to check the factor \[c^9 - 7d^9\] for further factorization. Upon examination, \[c^9 - 7 d^9\] does not simplify any more with integers, and thus it is classified as a prime polynomial.
Understanding the nature of prime polynomials is crucial, as it helps to conclude the factorization process confidently without missing any further simplifications.