Problem 56 - Algebra 2

Question

Find each product. Classify the result by number of terms. $$ (x+1)(x-1)(x+2) $$

Step-by-Step Solution

Answer

The product is \(x^3 + 2x^2 - x - 2\), which is classified as a polynomial with four terms (quadrinomial).

Step 1: Expand the First Two Binomials

Use the difference of squares formula, which states that \((a + b)(a - b) = a^2 - b^2\), to expand the first two binomials, \((x+1)(x-1)\). Here, \(a = x\) and \(b = 1\). Thus, the expansion is \(x^2 - 1^2\) which simplifies to \(x^2 - 1\).

Step 2: Multiply the Result by the Third Binomial

Now, multiply the result from the first step, \(x^2 - 1\), by the third binomial, \((x+2)\). This will require the distributive property, also known as the FOIL method for binomials, to multiply each term of \(x^2 - 1\) by each term of \(x+2\).

Step 3: Distribute Each Term and Combine Like Terms

When distributing, you get \((x^2 \times x) + (x^2 \times 2) - (1 \times x) - (1 \times 2)\). Simplifying this, you get \(x^3 + 2x^2 - x - 2\). There are no like terms to combine, so this is the final product.

Step 4: Classify the Result by Number of Terms

Count the number of terms in the final expression \(x^3 + 2x^2 - x - 2\). There are four terms, so the classification of this polynomial by the number of terms is a trinomial.

Key Concepts

Difference of Squares

Understanding the difference of squares is essential when multiplying certain polynomials. It is a specific case of a more general pattern in algebra where two terms are squared and subtracted from each other, described by the formula \(a^2 - b^2\). This can be factored into the product of two binomials \( (a+b)(a-b) \) which seemingly disappear when multiplied.

When faced with \( (x+1)(x-1) \), we can apply this concept. Here, \( a \) is \( x \) and \( b \) is \( 1 \), resulting in an expansion of \( x^2 - 1^2 = x^2 - 1 \). This simplifies the multiplication process, as recognizing this pattern can make calculations faster and more efficient. It's a neat trick that often appears in algebra, especially in factoring and simplifying expressions.

FOIL Method

The FOIL method is a useful tool for multiplying two binomials. FOIL stands for First, Outer, Inner, Last, referring to the order in which you multiply the terms in each binomial.

Let's take a closer look:

• First: Multiply the first terms in each binomial.
• Outer: Multiply the outermost terms in the product.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms in each binomial.

When you have a product like \(x^2 - 1)(x+2)\), although we don't have two binomials, we can still apply a similar approach by using the distributive property. Multiplying \(x^2\) by each term in \(x+2\) and then \( -1 \) by each term results in the desired expansion.

Distributive Property

The distributive property is also known as the distributive law of multiplication and division. It indicates that for any three numbers, a, b, and c, the equation \(a(b+c) = ab + ac\) holds true. This property allows us to multiply a sum or difference by distributing the value outside the parentheses to each term inside.

We utilized this property for expanding \( (x+1)(x-1)(x+2) \) when we multiplied the simplified expression \( x^2 - 1 \) by each term in the third binomial \( (x+2) \). By applying the distributive property, we distribute \( x^3 + 2x^2 \) and then \( -x - 2 \) to get the expanded form of the polynomial.

Combine Like Terms

After expanding polynomials, we often have to combine like terms to simplify the expression further. Like terms are terms in the expression that have the same variable raised to the same power. To combine them, we simply add or subtract their coefficients.

In the polynomial \( x^3 + 2x^2 - x - 2 \), the challenge to combine like terms was minimal because each term was unique in its variable and exponent. If we had terms like \( 3x^2 \) and \( 5x^2 \), we could combine them to get \( 8x^2 \). It's an essential part of simplifying expressions and ensuring that our polynomial is reduced to its most manageable form.