Problem 29 - Elementary Algebra

Question

For the following problems, factor the trinomials when possible. $$ x^{2}+13 x+42 $$

Step-by-Step Solution

Answer

Based on the step by step solution, provide the factored form of the trinomial \(x^2 + 13x + 42\). Answer: (x + 6)(x + 7)

Step 1: Identify the given trinomial

We are given the trinomial \(x^2 + 13x + 42\), and we need to factor it.

Step 2: Find two numbers that multiply to the constant term and add to the linear term's coefficient

We need to find two numbers whose product is the constant term (42) and whose sum is the coefficient of the linear term (13). After trying different pairs of factors of 42, we find that 6 and 7 satisfy the conditions: - \(6 \times 7 = 42\) - \(6 + 7 = 13\)

Step 3: Write the binomials

Using the numbers we found in step 2, write the trinomial as the product of two binomials: \((x + 6)(x + 7)\)

Step 4: Verify the factoring

To make sure our factoring is correct, we can expand the binomials to see if we get back to the original trinomial: \((x + 6)(x + 7) = x^2 + 7x + 6x + 42 = x^2 + 13x + 42\)

Step 5: Final Factored Form

The factored form of the trinomial \(x^2 + 13x + 42\) is: \((x + 6)(x + 7)\)

Key Concepts

Algebraic Expressions

Algebraic expressions are the backbone of algebra and are used to represent real-world problems in mathematical terms. An expression can consist of variables, coefficients, and arithmetic operations like addition, subtraction, multiplication, and division. In our example, the algebraic expression is a trinomial, denoted by three terms, which in this case are powers of a variable 'x' and constants. In the trinomial, \(x^2 + 13x + 42\), \(x^2\) is called a quadratic term, \(13x\) as the linear term, and 42 as the constant term.

Find Factors of Polynomials

Factoring polynomials is a crucial skill in algebra that involves breaking down a polynomial into simpler 'factor' expressions that, when multiplied together, return the original polynomial. The process, known as factoring by grouping, begins by identifying a pair of numbers that multiply to give the constant term (the last number) and add up to the coefficient of the linear term (the middle number).

As displayed in the step-by-step solution, for the trinomial \(x^2 + 13x + 42\), we found factors 6 and 7, which satisfy the conditions for factoring. This step is essential because the correct selection of numbers allows us to rewrite the trinomial as a product of two binomial expressions, simplifying the expression and making it more manageable for further calculations.

Binomial Products

Binomial products result from multiplying two binomials together. The process uses the distributive property, often referred to as the FOIL method, which stands for First, Outer, Inner, Last terms multiplication. This method simplifies the multiplication process.

In our exercise, the binomials \((x + 6)\) and \((x + 7)\) are multiplied to verify the factoring of the given trinomial. The property illustrates how two simple expressions, once factored correctly, interact to produce the original, more complex polynomial. Therefore, mastering binomial multiplication is critical, as it's used to both factor and expand polynomials in algebra.