Problem 154 - College Algebra

Question

Perform the indicated operation or operations. $$ (f(x))^{2}-2 f(x)+6, \text { where } f(x)-3 x-4 $$

Step-by-Step Solution

Answer

After performing the operations and simplifying, the result is \( 9x^{2} - 30x + 30 \).

Step 1: Substitute the Value of f(x)

Replace \( f(x) \) in the equation \( (f(x))^{2}-2 f(x)+6 \) with \( 3x - 4 \). This gives: \( ((3x-4)^{2}) - 2(3x-4) + 6 \)

Step 2: Expand the expression

When we expand \((3x-4)^{2}\), we get \(9x^{2} - 24x + 16\). Simplify the expression, we get \( 9x^{2} - 24x + 16 - 2(3x-4) + 6 \). Expand the entire expression to get: \( 9x^{2} - 24x + 16 - 6x + 8 + 6 \).

Step 3: Simplify the equation

Next, combine like terms, such as constants and x-terms: \( 9x^{2} - 30x + 30 \).

Key Concepts

Polynomial Expansion

Understanding polynomial expansion is crucial when dealing with algebraic expressions. Expansion involves converting a compacted expression, such as \( (a + b)^2 \) into a fully extended form, where all the products are written out. Taking the example from our exercise, \( ((3x - 4)^2) \) is a binomial squared, which requires us to multiply \( (3x - 4) \) by itself. This is where we apply the distributive property or the FOIL (First, Outer, Inner, Last) method.
When we expand \( (3x - 4)^2 \), we multiply each term in the first binomial by each term in the second binomial: \( (3x)(3x) = 9x^2 \) for the First terms, \( (3x)(-4) = -12x \) for the Outer terms, \( (-4)(3x) = -12x \) for the Inner terms, and \( (-4)(-4) = 16 \) for the Last terms. Gathering all these gives us \( 9x^2 - 12x - 12x + 16 \), which simplifies to \( 9x^2 - 24x + 16 \) as seen in Step 2 of the solution.

Practice for Mastery

Penciling down each product step when expanding can help avoid mistakes. Also, repetitive practice with different polynomials strengthens a student's ability to expand expressions quickly and accurately.

Substitution Method

The substitution method involves replacing a variable or an expression with a given number or another expression. It is a fundamental skill in algebra, particularly when working with functions. In our exercise, we are given \( f(x) \) as \( 3x - 4 \) and asked to perform operations on \( f(x)^2 - 2f(x) + 6 \).
As per Step 1, we substitute \( 3x - 4 \) for every occurrence of \( f(x) \) in the given expression, leading to \( ((3x - 4)^2) - 2(3x - 4) + 6 \) before proceeding with expansion. When using the substitution method, accuracy is paramount; replacing the function correctly ensures that the subsequent algebraic operations will lead to the correct result.

Verification Tip

After substituting, always revisit the original function or expression you replaced to confirm that the substitution was done correctly. This can prevent errors in the subsequent algebraic operations.

Combining Like Terms

Combining like terms is a process of simplifying algebraic expressions by adding or subtracting terms that are alike—in other words, terms that have the identical variable parts raised to the same power. A term is composed of its coefficient (the numerical part) and its variable part (the letter part).
In the final step of our problem, after expansion, we have terms like \( -24x \) and \( -6x \) which are like terms because they both contain the variable \( x \) raised to the first power. To combine them, we simply add or subtract their coefficients. The constants \( 16 \) and \( 6 \) are also like terms and can be combined by addition. In Step 3, we combine the like terms to arrive at \( 9x^2 - 30x + 30 \) by adding the coefficients of the like terms \( x \) and the constant terms respectively.

Effective Strategy

To avoid confusion, one might rewrite all like terms next to each other or underline them with the same color. This visual aid helps in identifying terms that can be combined more quickly and reduces the likelihood of combining unlike terms by mistake.