Problem 90 - Algebra 2

Question

Determine whether each polynomial is a perfect square trinomial. $$ x^{2}-10 x+16 $$

Step-by-Step Solution

Answer

No, it is not a perfect square trinomial.

Step 1: Identify the Structure of a Perfect Square Trinomial

A perfect square trinomial follows the form \((a+b)^2 = a^2 + 2ab + b^2\) or \((a-b)^2 = a^2 - 2ab + b^2\). To determine if a polynomial is a perfect square trinomial, it should match one of these forms.

Step 2: Verify the Squared Terms

Looking at the given polynomial \(x^2 - 10x + 16\), we identify \(a^2\) as \(x^2\), thus \(a = x\). We also need to check if 16 is a perfect square. Since \(4^2 = 16\), \(b = 4\).

Step 3: Check the Middle Term

For \(x^2 - 10x + 16 = (x-b)^2\), the middle term in the expansion should be \(-2ab = -2 \cdot x \cdot 4 = -8x\). However, the middle term provided in the polynomial is \(-10x\), which does not match \(-8x\).

Step 4: Conclusion

Since the middle term \(-10x\) does not satisfy the condition \(-2ab\) with \(a = x\) and \(b = 4\), \(x^2 - 10x + 16\) is not a perfect square trinomial.

Key Concepts

Polynomial Structure

In algebra, the structure of a polynomial is key to identifying patterns and forms like the perfect square trinomial. A perfect square trinomial is a specific type of quadratic polynomial that arises from the expansion of a binomial squared. It has the structure

• \((a+b)^2 = a^2 + 2ab + b^2\)
• \((a-b)^2 = a^2 - 2ab + b^2\)

These forms show that perfect square trinomials have specific relationships among their terms. This relationship is reflected in the coefficients and constants present in the polynomial. When analyzing the given polynomial \(x^2 - 10x + 16\), one should first consider if it fits into one of these structures. If it does, it can be simplified by factoring the trinomial back into a binomial squared.

Squared Terms

Squared terms in a polynomial provide a significant clue about its nature. In particular, the identification of squared terms can help verify whether the polynomial fits the perfect square trinomial form.
The squared terms are the first and last terms when the polynomial is expressed in standardized form \(ax^2 + bx + c\). In \(x^2 - 10x + 16\), we identify that the leading term \(x^2\) is a square, with \(a = x\).
Similarly, we check if the constant term \(16\) is also a square. Since the square root of 16 is 4, we have \(b = 4\). This potential match in squares is encouraging, but further verification is necessary to confirm that the polynomial is truly a perfect square trinomial.

Middle Term Verification

The middle term of a polynomial is crucial in verifying its status as a perfect square trinomial. This middle term results from the twice product of the two binomial components in the standard forms. Specifically, it should equal \(-2ab\) for

• \((a-b)^2\)
• \(2ab\) for \((a+b)^2\)

Let's examine \(x^2 - 10x + 16\). Given our potential values \(a = x\) and \(b = 4\), the expected middle term is \(-2 \times x \times 4 = -8x\). However, the polynomial has \(-10x\) as its middle term. This mismatch suggests that the polynomial does not meet the requirements to be classified as a perfect square trinomial, as the middle terms do not align.

Trinomial Factorization

Factorization is a process of breaking down a polynomial into simpler components, often binomials. For a polynomial to be a perfect square trinomial, it must factor into an identical binomial squared. However, when attempting to factor \(x^2 - 10x + 16\), a crucial observation must be made.
Since the polynomial's middle term does not fit the formula required for a perfect square trinomial, as established by our middle term verification, it cannot be factored as \((x-4)^2\) or any similar form. Therefore, this polynomial is not a perfect square trinomial and does not factor neatly into a square of a binomial.
Instead, other methods such as completing the square or using the quadratic formula might be necessary to explore further factorization, though they would not confirm it as a perfect square trinomial.