Problem 70 - College Algebra

Question

Factor completely, or state that the polynomial is prime. $$ 7 x^{4}-7 $$

Step-by-Step Solution

Answer

The completely factorized form of \(7x^{4} - 7\) is \(7(x-1)(x+1)(x^{2} + 1)\).

Step 1: Identify Common Factor

Notice that both terms in the expression \(7x^{4} - 7\) have common factor 7. To begin factoring, first factor out the common factor, which leads to the expression \(7(x^{4} - 1)\).

Step 2: Apply Difference of Squares

Observe that \(x^{4} - 1\) is a difference of squares, it can be factored using the identity \((a^2 - b^2) = (a-b)(a+b)\). If we let \(a = x^{2}\) and \(b = 1\), we have \(x^{4} - 1 = (x^{2}-1)(x^{2}+1)\). So the expression in Step 1 becomes \(7(x^{2} - 1)(x^{2} + 1)\).

Step 3: Further Factoring

Once again notice that \(x^{2} - 1\) is a difference of squares. If we let \(a = x\) and \(b = 1\), we have \(x^{2} - 1 = (x-1)(x+1)\). Thus, the expression in Step 2 becomes \(7(x-1)(x+1)(x^{2} + 1)\).

Key Concepts

Common Factor

Finding a common factor is often the first step in factoring polynomials. A common factor is a number or an algebraic expression that divides each term of the polynomial evenly. In the expression \(7x^{4} - 7\), both terms share the factor of 7. By factoring out the 7, the polynomial becomes simpler to work with: \(7(x^{4} - 1)\).

• Identify the greatest common factor (GCF) shared by all terms.
• Factor out the GCF, simplifying the expression.

This step is crucial because it reduces the complexity of the polynomial, making further factoring simpler. Always begin with this approach when faced with a polynomial! It sets you up for success.

Difference of Squares

After factoring out a common factor, the next method used is the difference of squares. This technique applies to expressions in the form \(a^2 - b^2\), which can be factored into \((a-b)(a+b)\). In the example \(x^{4} - 1\), we notice it can be rewritten as \((x^{2})^2 - 1^2\). Thus, it is a difference of squares.

• Recognize expressions like \(a^2 - b^2\) immediately.
• Apply the identity \((a-b)(a+b)\) to factor them.

Employing this technique allows you to break down what initially seems like a complex polynomial into simpler, more manageable terms. This step leverages the properties of squares to efficiently factor expressions.

Algebraic Expressions

Understanding algebraic expressions is key to successful factoring. An algebraic expression is a combination of numbers, variables, and operations. It can range from simple terms like \(x + 5\) to more complex polynomials. In the expression \(x^{4} - 1\), once it's recognized as a difference of squares, further factoring may reveal even simpler components.

• Analyze the structure of the algebraic expression thoroughly.
• Look for recognizable patterns like the difference of squares or distributive property.

In our exercise, after factoring \((x^{4} - 1)\) to \((x^{2} - 1)(x^{2} + 1)\), further breaking down is needed for \(x^{2} - 1\). This is again a difference of squares, yielding \((x-1)(x+1)\). Addressing each part one step at a time ensures a fully factored expression.