Question

s 27–32, fi nd the product of the binomials \((x-3)(x+2)(x+4)\)

Short answer

The product of the binomials \((x-3)(x+2)(x+4)\) is \(x^3 + 3x^2 - 3x - 24\).

Step-by-step solution

Multiply the first two binomials

Before anything else, multiply the first two binomials: \( (x - 3)(x + 2)\). Using the distributive property, it becomes: \(x*x + x*2 - 3*x - 3*2\), which simplifies to \(x^2 - x - 6\).

Multiply the result with the third binomial

Next, the answer from step 1 is multiplied with the third binomial: \( (x^2 - x - 6)(x + 4)\). Similarly, apply the distributive property which becomes: \(x^3 - x^2 - 6x + 4x^2 - 4x - 24\).

Simplify the result

Combine like terms to simplify the expression. In our case, it's as follows: \(x^3 - x^2 + 4x^2 - x - 6x + 4x - 24 = x^3 + 3x^2 - 3x - 24\).

Key concepts

Polynomial multiplication

Polynomial multiplication involves multiplying two or more polynomials together. This operation is critical in understanding and simplifying algebraic expressions. When performing this multiplication, each term from the first polynomial is multiplied by each term from the second polynomial. This process is repeated for any additional polynomials involved.

For example, when you multiply the binomials

• \((x - 3)\) and \((x + 2)\), each term in these binomials is multiplied by every term in the other binomial.
• This results in another polynomial, which can be simplified by combining like terms.

It's important to methodically apply the multiplication across all terms to make sure no terms are missed. Whether dealing with binomials, trinomials, or any higher-numbered polynomials, approach each multiplication step-by-step for clarity and accuracy.

Distributive property

The distributive property is a fundamental principle used in algebra to simplify expressions. It states that the product of a sum or difference is equal to the sum or difference of the products. It's written as \( a(b + c) = ab + ac \). This property is extensively used in polynomial multiplication.

• When applying to binomials like \((x - 3)(x + 2)\), distribute each term from the first binomial across the entire second binomial:
  \(x*(x + 2)\) + \(-3*(x + 2)\), creating new terms that are later combined.
• Use the distributive property to expand each pair, making sure each pair of terms is multiplied before moving to the next pair.

By using this method, every potential product is taken into account, ensuring that the final simplified expression is correct. The distributive property is a powerful and versatile tool in algebra that turns complicated polynomial expressions manageable.

Simplify expressions

Simplifying expressions is one of the core skills in algebra. After carrying out polynomial multiplication, it is crucial to combine like terms, ensuring the expression is as concise as possible.

• In the solution, after distributing and multiplying the terms \((x^2 - x - 6)(x + 4)\), individual terms are combined:
  \(x^3, -x^2,\) and \(4x^2\) are rearranged and added together.
• The like terms \(-x\) and \(-6x\) and \(4x\) are simplified to form a more straightforward expression:
  \(x^3 + 3x^2 - 3x - 24\).

By combining all like terms, we reduce the expression to its simplest form, which helps in solving problems more efficiently and understanding the relationship between variables. This process of simplification makes complex algebra problems more manageable and less intimidating.