Problem 15 - College Algebra Essentials

Question

In Exercises 15–58, find each product. $$(x+1)\left(x^{2}-x+1\right)$$

Step-by-Step Solution

Answer

The product of \((x+1)\) and \(\left(x^{2}-x+1\right)\) is \(x^{3}+1\)

Step 1: Distribution Step

Multiply each term in polynomial \(x+1\) with each term in polynomial \((x^{2}-x+1)\). So, the equation transforms into: \(x(x^{2}-x+1) + 1(x^{2}-x+1)\)

Step 2: Opening the Brackets

Open the brackets to get: \(x^{3}-x^{2}+x + x^{2}-x+1\)

Step 3: Combining Like Terms

Combine the like terms, that is those terms containing \(x^{2}\) and \(x\), to simplify the expression: \(x^{3} - x^{2}+ x^{2} + x - x + 1 \) which simplifies to \(x^{3} + 1\)

Key Concepts

Distributive Property

The distributive property is at the heart of multiplying polynomials. It allows us to break down a seemingly complex multiplication problem into smaller, more manageable pieces. This property states that multiplying a sum by a number is the same as multiplying each addend individually by the number and then adding the results. In the context of the given exercise, when you have \[ (x+1)(x^2-x+1) \] the distributive property helps by allowing the multiplication of each term in \(x+1\) by each term in \(x^2-x+1\). Here's a more detailed breakdown:

• Multiply \(x\) by each of the terms in \((x^2-x+1)\).
• Then, multiply \(1\) by each of the terms in \((x^2-x+1)\).

Bringing these results together fully employs the distributive property, transforming original expressions into simpler, solvable terms. This technique is fundamental for any polynomial multiplication.

Combining Like Terms

Once you have expanded the polynomial using the distributive property, you'll often find several 'like terms'. 'Like terms' are terms whose variables and their exponents are the same. Combining these terms is a crucial step in simplifying your expression.After using the distributive property on \[ x(x^2-x+1) + 1(x^2-x+1) \] , you end up with an expression:

• \(x^3 - x^2 + x + x^2 - x + 1\)

In this expression, identify terms that can be combined:

• Combine \(-x^2\) and \(+x^2\) to cancel out.
• Combine \(+x\) and \(-x\) to cancel out.

This step simplifies to \(x^3 + 1\), which is a much tidier outcome. Remember, combining like terms is not just about adding or subtracting numbers, but about recognizing patterns in polynomials.

Algebraic Expressions

Algebraic expressions, like the one in the given exercise \[ (x+1)(x^2-x+1) \] , are mathematical phrases that consist of variables, coefficients, and operators (like addition, subtraction, and multiplication). Understanding these elements is key to mastering polynomial multiplication.

Key Components of Algebraic Expressions:

• Variables: In our example, \(x\) is the variable. Variables stand for unknown or changeable values.
• Coefficients: These are numbers that multiply the variables. In the term \(-x^2\), the coefficient is \(-1\).
• Operators: Signs like \(+\) and \(-\) are operators that show the relationships between terms.

Handling these expressions involves applying operations systematically while maintaining the correct order of operations to ensure that solutions are accurate. Mastery in manipulating algebraic expressions is foundational for solving complex equations that students may face in calculus and beyond.