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

Ch. P - Fundamental Concepts of Algebra

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Ch. P - Fundamental Concepts of Algebra

문제 15

1장, 문제 15

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

검증된 단계별 안내

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.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.