Problem 96 - Introductory Algebra for College Students

Question

Factor $$3 x^{2}-15 x-42$$

Step-by-Step Solution

Answer

The factored form of the given equation is \((3x + 6) (x - 7)\)

Step 1: Multiply \(a\) and \(c\) in the given equation

Multiplication yields \(ac = 3(-42)=-126\)

Step 2: Finding two numbers

Find two numbers that will multiply to -126 and add up to -15. These two numbers are -21 and 6.

Step 3: Rewrite equation

With these numbers (6 and -21), rewrite the middle term: \(3x^2 - 21x + 6x -42\)

Step 4: Factoring by Grouping

Group the results into two groups (3x^2 - 21x) and (6x -42) then factor out a common factor from each group, the expression becomes \(3x(x-7) + 6(x - 7)\)

Step 5: Final Factoring

Since both groups include the factor \((x-7)\), it can be used as a common factor. The final factored equation is \((3x + 6) (x - 7)\)

Key Concepts

Polynomial Factorization

Understanding polynomial factorization is key to mastering algebra. It is a method of simplifying polynomials by expressing them as a product of their factors. By breaking down complex expressions into simpler components, we can solve equations more easily. In our exercise, our task is to factor the quadratic expression \(3x^2 - 15x - 42\). This means we need to express it as the product of two binomials or other simpler expressions.

To start, identify the coefficients from the expression: 3 is the coefficient of \(x^2\), -15 is the coefficient of \(x\), and -42 is the constant. The primary goal is to find two numbers that multiply to the product of the leading coefficient (3) and the constant term (-42) while also adding up to the middle coefficient (-15).

Once you successfully identify these numbers, you can split the linear term accordingly. This prepares you to apply various factoring techniques like grouping or using special identities, making polynomial factorization a powerful tool for simplifying algebraic expressions.

Factoring by Grouping

Factoring by grouping is a strategic method used when a polynomial expression can’t be readily factored through simple techniques. This involves rearranging and combining terms to uncover common factors, making it easier to simplify the equation.

In the given exercise, the quadratic expression \(3x^2 - 15x - 42\) is rewritten by splitting the middle term. This is achieved after finding two numbers, -21 and 6, that multiply to the product of the first and last coefficients (-126) and add up to the middle coefficient (-15). Thus, the expression is rewritten as \(3x^2 - 21x + 6x - 42\).

• First group terms: \((3x^2 - 21x) + (6x - 42)\).
• Identify common factors in each group: \(3x\) in the first, and 6 in the second group.
• Factor out these common factors: you get \(3x(x-7) + 6(x-7)\).

Notice how both groups reveal a common factor \((x-7)\), which simplifies further to \((3x + 6)(x - 7)\). This completes the factorization, making it a very efficient approach when dealing with complex expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (such as addition and multiplication) that act as the building blocks of algebra. They allow us to model real-world situations and solve various mathematical problems.

When working with algebraic expressions like \(3x^2 - 15x - 42\), it’s crucial to recognize terms, distinguish between coefficients and constants, and apply appropriate operations to simplify or solve equations. For effective manipulation, understanding how these parts interact is vital. Coefficients (like 3 and -15) adjust the rate of change or slope in linear components, while constants (-42) shift the overall graph or value in the expression.

• Terms are the separated components (e.g., \(3x^2, -15x, -42\)).
• The coefficients are the numeric factors (like 3 and -15) multiplying the variable terms.
• Constant terms are the standalone numbers without variables (e.g., -42).

Having a firm grasp on these concepts ensures that when you approach algebraic expressions, you can factor, rearrange, or simplify them systematically and accurately. This not only aids in solving equations but also enhances your understanding of the relationships between different elements within an expression.