Problem 81 - Algebra and Trigonometry

Question

Factor the expression completely. $$ \left(a^{2}+1\right)^{2}-7\left(a^{2}+1\right)+10 $$

Step-by-Step Solution

Answer

\((a-2)(a+2)(a-1)(a+1)\) is the completely factored form.

Step 1: Recognize a Quadratic Form

Notice that the expression \( \left(a^2+1\right)^2 - 7\left(a^2+1\right) + 10 \) is in a quadratic form regarding \( x = a^2 + 1 \). The expression can be rewritten as \( x^2 - 7x + 10 \).

Step 2: Factor the Quadratic Expression

Now, focus on factoring \( x^2 - 7x + 10 \). We need two numbers that multiply to 10 (the constant term) and add up to -7 (the linear coefficient). These numbers are -5 and -2, so the factored form will be \( (x - 5)(x - 2) \).

Step 3: Substitute Back the Expression for x

Now substitute \( x = a^2 + 1 \) back into the factored form. This gives us \( (a^2 + 1 - 5)(a^2 + 1 - 2) \), which simplifies to \( (a^2 - 4)(a^2 - 1) \).

Step 4: Further Factor Each Quadratic Term

Notice that both \( a^2 - 4 \) and \( a^2 - 1 \) are difference of squares. We can factor \( a^2 - 4 \) as \( (a - 2)(a + 2) \) and \( a^2 - 1 \) as \( (a - 1)(a + 1) \).

Step 5: Combine All Factors

Combine all the factors we found: \( (a - 2)(a + 2)(a - 1)(a + 1) \). This is the completely factored form of the original expression.

Key Concepts

Quadratic Form

A quadratic form is any expression that can be written in the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \). In the given exercise, we identify a particular structure within the expression \( \left(a^2+1\right)^2 - 7\left(a^2+1\right) + 10 \). Recognizing a quadratic form involves understanding that repeated structures, like \( a^2 + 1 \), can be treated as a single variable, \( x \). This allows us to rewrite the expression as \( x^2 - 7x + 10 \). Using this technique simplifies the process of factoring, as we can now apply familiar methods to the new quadratic form. Identifying these forms is crucial because it transforms complex expressions into simpler forms, making further computations easier.

Difference of Squares

The difference of squares is a special algebraic identity that is very helpful in factoring: \( a^2 - b^2 = (a-b)(a+b) \). This identity applies when you have two terms, each a perfect square, separated by a subtraction sign. In our exercise, after substituting and partially factoring, we are left with \( (a^2 - 4)(a^2 - 1) \). Both of these are in a difference of squares form:

• \( a^2 - 4 = (a^2 - 2^2) = (a-2)(a+2) \)
• \( a^2 - 1 = (a^2 - 1^2) = (a-1)(a+1) \)

Recognizing and applying the difference of squares can simplify factoring significantly. It breaks down the expression into basic binomials that are easier to handle.

Quadratic Expression

Factoring a quadratic expression involves finding two binomial expressions that multiply to give the original quadratic. Specifically, for the quadratic \( x^2 - 7x + 10 \), we looked for numbers that multiply to 10 (the constant) and add to -7 (the coefficient of \( x \)). These numbers are -5 and -2, leading to the factorization \( (x-5)(x-2) \). Factoring quadratic expressions is one of the foundational skills in algebra, integral to simplifying and solving equations. Mastering this skill involves practice in recognizing patterns and calculating combinations that satisfy the condition of both multiplying and adding to given parameters.