Problem 66 - Elementary and Intermediate Algebra

Question

Factor. If an expression is prime, so indicate. $$ 170 h^{2}-210 h-260 $$

Step-by-Step Solution

Answer

Factor is \(10(17h^2 - 21h - 26)\) with \(17h^2 - 21h - 26\) being prime.

Step 1: Identify Common Factors

First, determine if there's any greatest common factor (GCF) for the coefficients 170, -210, and -260. Notice that each of these numbers is divisible by 10. Factoring out 10 from the whole expression gives us: \[ 10(17h^2 - 21h - 26) \]

Step 2: Consider Factoring the Quadratic

Now, focus on factoring the quadratic expression inside the parenthesis: \(17h^2 - 21h - 26\). The coefficients aren't easily factorable unless they relate to a special condition, further factorization in integers might not be possible.

Step 3: Verify if Expression is Prime

Attempt to find two numbers that can multiply to \(-17 \times 26 = -442\) and add up to \(-21\). This value doesn't break down into simple integer factors, confirming no further factoring is likely unless using irrational numbers.

Step 4: Confirm the Expression is Prime

Since no integer factors satisfy the conditions, the expression inside the parenthesis, \(17h^2 - 21h - 26\), is a prime polynomial. It cannot be factored further using integers.

Key Concepts

Understanding the Greatest Common Factor

The greatest common factor (GCF) is a key concept in polynomial factorization. It helps simplify expressions by dividing each term of the polynomial by the largest number that divides all terms evenly.
To find the GCF, follow these steps:

• List the factors of each coefficient.
• Identify the largest common factor shared by all coefficients.

In the expression given, the coefficients are 170, -210, and -260. Each of these numbers has 10 as a common factor.
By factoring out the GCF of 10, the polynomial simplifies, allowing for easier manipulation of the remaining terms: \[ 170h^2 - 210h - 260 = 10(17h^2 - 21h - 26) \]
This process reduces complexity and paves the way for potential further factorization or confirmation of prime status.

Exploring Prime Polynomials

A prime polynomial is one that cannot be factored further over the integers. This means no pair of polynomials with integer coefficients multiply together to form the original polynomial, except for the simplest combinations like 1 and itself.
When determining if a polynomial like the quadratic \( 17h^2 - 21h - 26 \) is prime, a few key checks are necessary:

• Check if it can be factored into products of polynomials with smaller integer coefficients.
• Attempt to decompose it into linear factors using real or rational numbers.

In this case, simplifying the polynomial inside the parentheses proved impossible with integer solutions since the product and sum conditions required to break down the middle term \( -21h \) into simpler integer components were not met.
As a result, the expression \( 17h^2 - 21h - 26 \) is classified as a prime polynomial in relation to integer factorization.

Factoring Quadratic Expressions

Quadratic expressions are in the standard form \( ax^2 + bx + c \). Factoring these polynomials is a common exercise in algebra and involves breaking down the expression into the product of two binomials.
For successful factorization, a few strategies might help:

• Seek two numbers that multiply to \( a imes c \) (the product of the first and last coefficient).
• These numbers should also add up to \( b \), the middle coefficient.

In the example expression \( 17h^2 - 21h - 26 \), the goal is to find such two numbers. In this scenario, the multiplication yields \( -442 \) and the sum requires \( -21 \).
However, no integer pairs meet both criteria, implying that further simple simplifications using integers aren't possible.
Therefore, if a quadratic cannot be factored this way, it may either be irreducible on integer level, or require more advanced methods, confirming its prime status in simpler contexts.