Problem 106 - Algebra and Trigonometry

Question

Factor Completely. $$7 x^{4}+34 x^{2}-5$$

Step-by-Step Solution

Answer

The completely factored form of the expression is \( (\sqrt{7}x - 1)(\sqrt{7}x + 1)(x^2 + 5)\).

Step 1: Identifying the Quadratic Form

Recognize that the expression is in the form of \( ax^4 + bx^2 + c \). The first step to solve this task is to substitute \(x^2 = m\), then the task becomes \(7m^2 + 34m - 5\). Now, we have a quadratic expression with respect to \(m\).

Step 2: Factoring the Expression

Next, one should apply the technique for factoring quadratic trinomials which is to look for two numbers that multiply to \(ac = -35 \times 7 = -245\) (product of the coefficients of \(m^2\) and the constant term) and add up to \(b = 34\) (coefficient of \(m\)). After a little trial and error, we find that these numbers are 35 and -7. Using these numbers, the quadratic expression can be expressed as follows: \(7m^2 + 35m - 7m - 5 = 7m(m + 5) - 1(m + 5)\). Factoring out the shared terms gives \( (7m - 1)(m + 5)\).

Step 3: Substitute back

Now, substitute \(m\) back with \(x^2\). Doing so gives us \( (7x^2 - 1)(x^2 + 5)\). Now, we could stop here but, \((7x^2 - 1)\) can be factored further as \((\sqrt{7}x - 1)(\sqrt{7}x + 1)\).

Key Concepts

Quadratic Expression

A quadratic expression is an algebraic expression of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. It's called quadratic because the highest power of the variable is 2. In the problem provided, the expression \( 7x^4 + 34x^2 - 5 \) is in a form similar to a quadratic when we consider powers of \( x^2 \) as the variable.
Understanding quadratic expressions is fundamental in algebra because they appear frequently in various types of equations and real-world problems. Recognizing the structure of a quadratic expression allows us to apply systematic methods for solving or factoring it. This is why the substitution step is so crucial; it transforms a complex expression into a familiar quadratic form, easing the process of manipulation.

• Identify the standard quadratic form \( ax^2 + bx + c \).
• Use substitutions to reduce higher-degree polynomials to quadratics, when possible.
• Employ this recognition in solving and factoring tasks.

Trinomials

Trinomials are algebraic expressions composed of three terms. In a quadratic form, they're typically written as \( ax^2 + bx + c \). Factoring trinomials involves finding two binomials whose product equals the trinomial.
The expression from the exercise, after substitution, simplifies to the trinomial \( 7m^2 + 34m - 5 \). An essential part of solving such problems is understanding that each trinomial can potentially be expressed as the product of two binomials, as we did with \( (7m - 1)(m + 5) \).
In algebra, factoring trinomials is a powerful technique used to simplify equations and solve them. Recognizing the pattern of a trinomial makes it easier to solve quadratic equations and is a vital skill in algebra and pre-calculus. By breaking down a complex expression into simpler multiplicative components, you can handle much more complex equations effectively.

• Recognize the trinomial pattern as \( ax^2 + bx + c \).
• Factor by grouping terms, when applicable.
• Transform complex polynomials into manageable components.

Substitution Method

The substitution method is a strategic technique in algebra used for simplifying expressions and equations. It involves replacing one variable or expression with another equivalent to transform a complex problem into a simpler or more familiar form. In the original exercise, we used substitution to transform the quartic polynomial \( 7x^4 + 34x^2 - 5 \) into a quadratic \( 7m^2 + 34m - 5 \) by letting \( m = x^2 \).
This technique is particularly useful when dealing with polynomials and equations involving higher powers. By substituting parts of the expression, we create an easier scenario, often converting into a more familiar quadratic or linear form that we can solve directly. When factoring, substitution often unveils a straightforward path to simplifying expressions or solving equations.

• Identify the part of the expression that can be substituted.
• Choose an appropriate substitution to simplify the problem.
• Substitute back after simplification to find the solution in the original terms.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. Understanding and manipulating algebraic expressions are key skills in algebra. These expressions represent real-world quantities and relationships, allowing us to model and solve problems.
Algebraic expressions range from simple linear expressions like \( 2x + 3 \) to complex polynomials such as \( 7x^4 + 34x^2 - 5 \). The ability to factor these expressions, as demonstrated in the solution, is pivotal for solving equations and understanding deeper algebraic structures.
Each term in an expression corresponds to a specific part of the problem it represents. In the context of polynomials, being able to break down or combine these expressions simplifies problem-solving considerably. This involves recognizing patterns, such as those found in trinomials or quadratics, and applying appropriate techniques like substitution or factoring.

• Recognize and correctly interpret the components of algebraic expressions.
• Apply transformations to these expressions for simplification purposes.
• Use algebraic expressions to model and solve real-world problems.