Problem 41 - Algebra 2

Question

Write each function in standard form. $$ y=(x+1)(x-4)(3-2 x) $$

Step-by-Step Solution

Answer

The standard form of the given function is \(y = -2x^3 + 5x^2 + 6x -12\).

Step 1: Distribute the First Two Factors

Multiply the first two factors, \(x+1)\) and \(x-4)\), using the FOIL (First, Outer, Inner, Last) method. This gives us a quadratic expression.

Step 2: Multiply the Resulting Quadratic by the Third Factor

Now, multiply the quadratic expression obtained from Step 1 by the third factor, \(3 - 2x)\), through distribution.

Step 3: Combine Like Terms

Combine like terms to write the final expression in standard form, which is \(ax^2 + bx + c)\) where \(a\), \(b\), and \(c\) are real numbers.

Key Concepts

FOIL Method

The FOIL method is a technique used to multiply two binomials. FOIL is an acronym that stands for First, Outer, Inner, Last, referring to the order in which you multiply the terms in each binomial. Everything hinges on the distribution property, which tells us how to handle multiplication across parentheses.

To demonstrate, consider the expression \( (x+1)(x-4) \). Apply the FOIL method as follows:

• First: Multiply the first terms in each binomial, \(x * x = x^2\).
• Outer: Multiply the outer terms, \(x * -4 = -4x\).
• Inner: Multiply the inner terms, \(1 * x = x\).
• Last: Multiply the last terms, \(1 * -4 = -4\).

The result of these multiplications is combined to create a quadratic expression, \(x^2 - 4x + x - 4\), which simplifies to \(x^2 - 3x - 4\). This is an intermediate step in our problem before multiplying by the third factor.

Quadratic Expressions

Quadratic expressions are polynomial expressions of degree 2, generally represented as \(ax^2 + bx + c\) where \(a\), \(b\), and \(c\) are real numbers and \(a \eq 0\). The process of expanding binomials, like in the FOIL method, often results in quadratic expressions. These expressions form parabolas when graphed and are central to a multitude of algebraic problems.

In the context of the given exercise, multiplying two binomials by the FOIL method generates a quadratic expression. This expression is then further expanded when it’s multiplied by the third factor, \(3 - 2x\), to eventually reach the standard form. Knowing how to manipulate quadratic expressions is crucial for solving and graphing quadratic equations.

Combining Like Terms

Combining like terms is a fundamental technique in algebra used to simplify expressions. Like terms are terms within an expression that have the same variables raised to the same powers, though they can have different coefficients. The process of combining these is effectively just an application of basic addition or subtraction.

After the multiplication steps in the initial problem, you are left with terms that may contain the variable \(x\) to various powers or be constant. To reach standard form, group all the \(x^2\) terms, the \(x\) terms, and the constant terms. Summarize them to achieve the final expression in the standard form of \(ax^2 + bx + c\). For instance, if we had \(x^2 - 4x + x - 4\) and then multiplied by \(3 - 2x\), we would follow these steps to combine like terms and finally simplify our quadratic to its standard form.