Problem 50 - Elementary Algebra

Question

Use a pattern to factor. Check. Identify any prime polynomials. $$ h^{2}+4 h+4 $$

Step-by-Step Solution

Answer

\((h^2 + 4h + 4)\) factors to \((h + 2)^2\), and it is not a prime polynomial.

Step 1 - Identify the quadratic form

Notice that the polynomial is a quadratic trinomial in the form of \(ax^2 + bx + c\). Here, \(a = 1\), \(b = 4\), and \(c = 4\).

Step 2 - Check for a perfect square trinomial

A perfect square trinomial takes the form \((x + k)^2 = x^2 + 2kx + k^2\). Compare this form to \(h^{2} + 4h + 4\). Here, \(x = h\) and we need to check if the middle term \(4h\) can be written as \(2 \times h \times k\) and the last term \(4\) as \(k^2\).

Step 3 - Find the value of \(k\)

We solve \(2hk = 4h\) to find \(k\). By setting \(2h \times k = 4h\), we get \(k = 2\). Also, \(k^2 = 4\) is true for \(k = 2\). Hence, \(h^2 + 4h + 4\) can be factored as \((h + 2)^2\).

Step 4 - Verify the factorization

Expand \((h + 2)^2\) to confirm the factorization: \((h + 2)(h + 2) = h^2 + 2h + 2h + 4 = h^2 + 4h + 4\). This matches the original polynomial, so \((h + 2)^2\) is correct.

Step 5 - Check if it's a prime polynomial

A prime polynomial cannot be factored further. Since \(h^2 + 4h + 4\) has been factored as \((h + 2)^2\), it is not a prime polynomial.

Key Concepts

perfect square trinomial

A perfect square trinomial simplifies factoring. It follows a specific pattern. ewline ewline A perfect square trinomial has the form ewline ewline \((x + k)^2 = x^2 + 2kx + k^2\) ewline ewline Recognizing this pattern is crucial. When factoring \(h^2 + 4h + 4\), compare it to \((x + k)^2\). ewline Here, \(x = h\) and you compare terms to find \(k\). ewline Notice, 4h fits as \(2hk\). Therefore, \(k\) is determined as 2 because \(2h \times 2 = 4h\). ewline The last term matches \(k^2\). So \(k = 2\) because \(2^2 = 4\). This formation confirms a perfect square trinomial.

quadratic form

Quadratic form is represented by the equation \(ax^2 + bx + c\). ewline In this problem, \(h^2 + 4h + 4\), each coefficient is clear: ewline \(a = 1\), \(b = 4\), and \(c = 4\). ewline Recognizing the quadratic nature helps in identifying a pattern for factoring. Examining how these coefficients relate is essential. ewline When \(a = 1\), your trinomial might be a perfect square. This makes it easier to use simplification methods.

factoring polynomials

Factoring polynomials means breaking them into products of simpler polynomials. ewline For instance, factoring \(h^2 + 4h + 4\) results in \((h + 2)^2\). ewline To factor, notice the quadratic form first. Then use patterns or methods like grouping or the quadratic formula. ewline ewline Here's a common approach:

• Identify coefficients
• Check for special forms like perfect squares
• Apply patterns accordingly

ewline Verify the factorization by expanding back to the original polynomial to ensure accuracy.

prime polynomials

Prime polynomials cannot be factored further using real numbers. ewline In this exercise, checking whether \(h^2 + 4h + 4\) is prime involves factoring initially. ewline After factoring to \((h + 2)^2\), recognize it's not prime as it can be simplified. ewline If you can't simplify more, then the polynomial is prime. Here, the polynomial \(h^2 + 4h + 4\) breaks down into simpler factors, proving it's not prime. Confirming it's not prime ensures comprehension and correct completion of factoring tasks.