In Exercises 15–58, find each product. (x+1)(x2−x+1)

Ch. P - Fundamental Concepts of Algebra

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

All textbooks

Blitzer 8th Edition

Ch. P - Fundamental Concepts of Algebra

Problem 15

Chapter 1, Problem 15

In Exercises 15–58, find each product. (x+1)(x²−x+1)

Verified step by step guidance

Step 1: Recognize that the problem involves multiplying two polynomials: $(x + 1)$ and $(x^2 - x + 1)$. This requires the distributive property (also known as the FOIL method for binomials).

Step 2: Distribute the first term of $(x + 1)$, which is $x$, to each term in $(x^2 - x + 1)$. This gives: $x \cdot x^2 + x \cdot (-x) + x \cdot 1$.

Step 3: Simplify the terms from Step 2: $x \cdot x^2 = x^3$, $x \cdot (-x) = -x^2$, and $x \cdot 1 = x$. Combine these to get $x^3 - x^2 + x$.

Step 4: Distribute the second term of $(x + 1)$, which is $1$, to each term in $(x^2 - x + 1)$. This gives: $1 \cdot x^2 + 1 \cdot (-x) + 1 \cdot 1$.

Step 5: Simplify the terms from Step 4: $1 \cdot x^2 = x^2$, $1 \cdot (-x) = -x$, and $1 \cdot 1 = 1$. Combine these to get $x^2 - x + 1$. Finally, add the results from Step 3 and Step 5 to combine like terms.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polynomial Multiplication

Polynomial multiplication involves distributing each term of one polynomial to every term of another polynomial. In this case, we will apply the distributive property to multiply the binomial (x + 1) with the trinomial (x^2 - x + 1), ensuring that each term in the first polynomial is multiplied by each term in the second.

Combining Like Terms

After multiplying polynomials, the next step is to combine like terms, which are terms that have the same variable raised to the same power. This process simplifies the expression into a standard polynomial form, making it easier to analyze or further manipulate.

Standard Form of a Polynomial

The standard form of a polynomial is expressed in descending order of the powers of the variable. For example, a polynomial like 2x^3 + 3x^2 + x + 5 is in standard form. Writing the final product in standard form helps in understanding the polynomial's degree and leading coefficient.

Recommended video:

Guided course

05:16

Standard Form of Polynomials

Related Practice

Textbook Question

Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. √27

views

Textbook Question

Use the product rule to simplify the expressions in Exercises 13–22. In Exercises 17–22, assume that variables represent nonnegative real numbers. $the square root of 125 x squared$

views

Textbook Question

In Exercises 11–16, factor by grouping. x³+6x²−2x−12

Textbook Question

In Exercises 7–14, simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. (x²−14x+49)/(x²−49)

views

Textbook Question

In Exercises 1–16, evaluate each algebraic expression for the given value or values of the variable(s). (2x+3y)/(x+1), for x=-2 and y=4

Textbook Question

Multiply or divide as indicated. $\frac{6x + 9}{3x - 15}\) $\cdot$ \(\frac{x - 5}{4x + 6}$

views