Problem 87 - Algebra A Combined Function

Question

See the Concept Check in this section. Do the terms of \(4 x^{2}+19 x+12\) have a common factor (other than 1\() ?\)

Step-by-Step Solution

Answer

No, the terms do not have a common factor other than 1.

Step 1: Identify the terms

The expression is \(4x^2 + 19x + 12\). It consists of three terms: the first term is \(4x^2\), the second term is \(19x\), and the third term is \(12\).

Step 2: Find the factors of each coefficient

The coefficient of the first term is 4, which factors into \(2 \times 2\). The coefficient of the second term is 19, which is a prime number with factors 1 and 19. The constant term is 12, which factors into \(3 \times 2 \times 2\).

Step 3: Determine common factors

Compare the factors of the coefficients: 4 has the factors \(2, 2\); 19 has the factors \(1, 19\) (since it's a prime number); and 12 has the factors \(2, 2, 3\). The only common factor among all coefficients is 1.

Step 4: Verify the common factor conclusion

Since the only common factor among the coefficients 4, 19, and 12 is 1, and 1 is typically not considered a common factor for this type of question, the terms do not have a common factor other than 1.

Key Concepts

Common Factor

When we talk about common factors in a polynomial, we refer to numbers that evenly divide all the coefficients of the polynomial's terms. In the expression \(4x^2 + 19x + 12\), our task is to determine if these terms have any common factors other than 1.

• The term \(4x^2\) has a coefficient of 4.
• The term \(19x\) has a coefficient of 19.
• The term 12 is a constant and has itself as a coefficient.

By listing each term's factors:

• 4 factors into \(2 \times 2\).
• 19 is only divisible by 1 and 19.
• 12 factors into \(2 \times 2 \times 3\).

To find a common factor, we look for numbers appearing in each list of factors. In this case, the only number that fits this criterion is 1. Thus, the terms of this polynomial do not share a common factor other than 1.

Prime Number

A prime number is a natural number greater than 1, which can only be divided by 1 and itself. This property makes prime numbers essential building blocks in number theory. In the context of the polynomial \(4x^2 + 19x + 12\), we focus on the coefficient of the second term, which is 19.
19 fits the definition of a prime number because it has no divisors other than 1 and 19. This means there are no smaller values other than these that 19 can be divided by without a remainder.
Understanding prime numbers helps us quickly conclude that such numbers can be excluded from shared factors unless predominantly present in all terms or coefficients. Here, it assists in asserting that no common factors other than 1 exist among the terms, simplifying the factorization process.

Coefficients

In polynomial expressions, coefficients are the numerical parts that multiply each term's variables. For the polynomial \(4x^2 + 19x + 12\), the coefficients are 4, 19, and 12. Each one influences the polynomial's factorization in distinct ways.

• For \(4x^2\), the coefficient is 4, which can be broken down into its factors \(2 \times 2\).
• The coefficient 19 in \(19x\), as a prime number, contributes itself as it can't be decomposed further.
• Finally, 12 is the coefficient for the constant part, decomposing into \(3 \times 2 \times 2\).

Coefficients are crucial in determining a polynomial's factorization, as they dictate potential shared factors between terms. In this polynomial, analyzing coefficients helps us deduce that no common number exists among them, meaning the expression doesn't benefit from simplification by a common factor.