Problem 21 - Introductory Algebra for College Students

Question

Use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$8 x^{2}+33 x+4$$

Step-by-Step Solution

Answer

The trinomial \(8x^{2}+33x+4\) is prime and cannot be factorized.

Step 1: Factorize the trinomial

Here we have to find two numbers that multiply to \(8*4=32\), but add up to \(33\). Unfortunately, there are no such numbers. Thus, the trinomial is prime and cannot be factorized.

Step 2: Verify non-factorization

One way to verify if a trinomial is prime is to use the discriminant method, where we calculate \(b^{2}-4ac\). In this case, a coefficient is \(8\), b coefficient is \(33\), and c coefficient is \(4\). For our case, the discriminant method results in \(33^{2}-4*8*4 = 961\), which is not a perfect square. Thus, it confirms that the trinomial is prime and can’t be factored with integers.

Key Concepts

Prime Polynomial

A prime polynomial is an interesting concept in algebra. It refers to a polynomial that cannot be factored further with integer coefficients. Imagine it like a prime number, which can't be divided by any number other than 1 and itself. In our exercise with the trinomial \(8x^2 + 33x + 4\), we tried to factor it but found no suitable pairs of numbers that multiply to 32 (product of the constants) and add to 33 (the linear coefficient). As a result, this polynomial is considered prime.

• If a polynomial is prime, it can't be broken into simpler polynomial factors.
• The prime nature was confirmed through the quadratic discriminant showing no perfect square.

Every polynomial has its own uniqueness, and sometimes it refuses to be factored, like a true prime polynomial.

FOIL Multiplication

FOIL multiplication is an essential method for multiplying two binomials. It's named FOIL for an easy-to-remember process:

• F stands for multiplying the **First** terms of each binomial.
• O stands for the **Outside** terms multiplication.
• I stands for **Inside** terms multiplication.
• L stands for **Last** terms multiplication.

By combining these products, you gain a thorough understanding of how to expand binomials into trinomials or larger polynomial expressions. While it wasn't directly used in determining the primality of our trinomial, FOIL becomes a handy tool to check work when factoring polynomials. It assures that the decomposition into factors, if possible, accurately gives back the original expression when reorganized.

Quadratic Discriminant

The quadratic discriminant is a valuable tool that gives insights into the nature of the solutions of a quadratic equation. It is found using the formula \(b^2 - 4ac\), where \(a\), \(b\), and \(c\) are coefficients of the quadratic expression \(ax^2 + bx + c\). For the trinomial \(8x^2 + 33x + 4\), we calculated:\[33^2 - 4 \times 8 \times 4 = 961\]Since 961 is not a perfect square, it indicates that the trinomial has no real roots. Here's why it's significant:

• If the discriminant is a perfect square, the polynomial can be factored with rational numbers.
• A non-perfect square discriminant means it can't be factored using integers, confirming it's prime.

The utility of the discriminant in our exercise helped in confirming that our polynomial is indeed prime, by making evident that no integer-based factorization exists.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operators like addition and multiplication. They form the backbone of algebra and are used to represent mathematical relationships and operations.The given expression \(8x^2 + 33x + 4\) is an example of a quadratic algebraic expression. It includes:

• **Coefficient**: The numerical factor of a term. For instance, 8 in \(8x^2\).
• **Variable**: The letter that represents a number, like \(x\) in this polynomial.
• **Degree**: The highest exponent of the variable, here it's 2, making it a quadratic expression.

Understanding these elements is pivotal in operations like factoring, expanding, and simplifying expressions. The process of factorization involves breaking down a complex algebraic expression into simpler terms, unless it is prime, just like in our exercise where such factorization wasn't feasible.