Problem 64 - Algebra 1: Concepts and Skills

Question

Find the product \((x-1)\left(2 x^{2}+x+1\right)\) $$ \text { (A) } 2 x^{3}-3 x^{2}-1 $$ $$ \text { (B) } 2 x^{3}-x^{2}-2 x-1 $$ $$ \text { (C) } 2 x^{3}-x^{2}-1 $$ $$ \text { (D) } 2 x^{3}+3 x^{2}+2 x+1 $$

Step-by-Step Solution

Answer

(B) \(2x^{3}-x^{2}-x-1\)

Step 1: Distribute the outer terms

We will first multiply each term in the polynomial \( (x-1) \) by the outermost term in the polynomial \( \left(2 x^{2}+x+1\right) \). So we get: \(2x^{3}+x^{2}+x\).

Step 2: Distribute the inner terms

We will do the same process with the inner terms being multiplied by each term in \( \left(2 x^{2}+x+1\right) \) . So we get: \(-2x^{2}-x-1\).

Step 3: Combine like terms

Now we need to combine like terms, which in this case are \( x^{2} \) terms and \( x \) terms. When we combine these terms, we get: \(2x^{3}-x^{2}-x-1\).

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables (like x or y), and operation symbols (such as +, -, *, and /). It’s a way to represent numbers in a general form. For example, in the expression (2x^{2} + x + 1), 2x^{2} is a term that tells us there are 2 units of x squared, x implies there is 1 unit of x, and 1 is the constant term.

In the product (2x^{3}-x^{2}-x-1), we see a polynomial, which is basically a type of algebraic expression where variables are raised to whole number exponents and the operations can be addition, subtraction, and multiplication. To find such products accurately, you need to understand how variables interact with each other and with numbers through multiplication.

Combining Like Terms

When working with algebraic expressions, 'combining like terms' is a method used to simplify an equation or expression. Like terms are terms whose variables (and their exponents) are the same. For instance, x^{2} and -2x^{2} are like terms since they both contain x squared. In contrast, x^{2} and x are not like terms because the exponents on x are different.

In our step-by-step solution, combining like terms is essential in Step 3, where we combine terms involving x^{2} and x to simplify the polynomial to 2x^{3}-x^{2}-x-1. Making sure to correctly identify and combine like terms is crucial for correctly solving the multiplication of polynomials and arriving at the right answer.

Distributive Property

The distributive property is a fundamental algebraic property used in simplifying equations and multiplying terms. It states that for any numbers or expressions a, b, and c, the expression a(b + c) equals ab + ac. In other words, you distribute the multiplication of a across the addition inside the parentheses.

In our problem, we utilize the distributive property twice — first to multiply (x-1) by 2x^{2} which is in the polynomial (2x^{2} + x + 1), and again to multiply (x-1) by the remaining terms in the polynomial. These consecutive applications of the distributive property are key to expanding the given product into a polynomial form before combining like terms to reach the final simplified version.