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Magazine Chapter 6: Problem 8
Identify each of the following as a perfect-square trinomial, a difference of two squares, a prime polynomial, or none of these. $$ t^{2}-6 t+8 $$
Short Answer
Expert verified
None of these
Step by step solution
01
Identify the Polynomial
Given the polynomial: t^2 - 6t + 8
02
Check for Perfect-Square Trinomial
A perfect-square trinomial takes the form a^2 + 2 ab + b^2. Let's see if we can write t^2 - 6t + 8 into that form.In this case, a^2 = t^2, so a = t. Checking for b, ... compare the term 2ab with -6t to find b = -3 and so b^2 = 9.Clearly the constant term is 8 which do not match.So, it's not a perfect-square trinomial.
03
Check for Difference of Squares
A difference of squares takes the form: a^2 - b^2. Given polynomial does not have this form as it has three terms instead of two.So, it's not the difference of squares.
04
Check for Prime Polynomial
A prime polynomial cannot be factored into simpler polynomials. Let's try factoring t^2 - 6t + 8 using factoring techniques.Using factorization method:Look for two numbers whose product is 8 and sum is -6. Numbers -4 and -2 fit, leading to factorization:(t-4)(t-2) =0 Hence, the given polynomial t^2 - 6t + 8 is factorable, so it鈥檚 not a prime polynomial.
05
Conclude the Type of Polynomial
Since t^2 - 6t + 8 is neither a perfect-square trinomial, nor a difference of squares and is not prime because it can be factored, none of these categories apply.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Perfect-Square Trinomial
A perfect-square trinomial is a special type of quadratic expression. It takes the form \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\).
- The first term, \(a^2\), is always a square of some term \(a\).
- The second term is twice the product of \(a\) and another term \(b\).
- The last term, \(b^2\), is the square of term \(b\).
Difference of Squares
The difference of squares is another recognizable polynomial pattern. It takes the form \(a^2 - b^2\) and can be factored into \((a + b)(a - b)\). This form only consists of two terms: \(a^2\) and \(-b^2\).
- The first term, \(a^2\), is a square of some term \(a\).
- The second term, \(-b^2\), is a negative square of some term \(b\).
Prime Polynomial
A prime polynomial is one that cannot be factored into the product of two or more non-constant polynomials. These polynomials are like prime numbers in arithmetic because they have no factors other than 1 and themselves. To determine if a polynomial is prime, one can try factoring it. For instance, \( x^2 + x + 1 \) is a prime polynomial because it cannot be factored further. In the exercise, the polynomial \( t^2 - 6t + 8 \) is not prime because it can be factored into \((t - 4)(t - 2)\). Hence, \( t^2 - 6t + 8 \) is not a prime polynomial.
Factoring Polynomials
Factoring polynomials is a process of breaking down a polynomial into simpler terms (factors) that, when multiplied together, give back the original polynomial. This is useful for solving polynomial equations and simplifying expressions.
- Look for common factors: Identify and factor out the greatest common factor (GCF).
- Recognize patterns: Identify patterns like perfect-square trinomials, differences of squares, and other recognizable factors.
- Use the factorization method: For trinomials, find two numbers that multiply to give the constant term and add to give the coefficient of the middle term.
