Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. 2x³+x+2; x+1

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Identify the polynomials involved: the first polynomial is \(2x^{3} + x + 2\) and the second polynomial is \(x + 1\).

Rewrite the divisor \(x + 1\) in the form \(x - r\) to find the value of \(r\). Since \(x + 1 = x - (-1)\), we have \(r = -1\).

Apply the Factor Theorem by evaluating the first polynomial at \(x = r = -1\). Substitute \(-1\) into \(2x^{3} + x + 2\) to check if the result is zero.

If the result is zero, then \(x + 1\) is a factor of the first polynomial. If not, proceed to use synthetic division to divide \(2x^{3} + x + 2\) by \(x + 1\).

Set up synthetic division using \(r = -1\) and the coefficients of the first polynomial (note the missing \(x^{2}\) term has coefficient 0). Perform the division step-by-step to find the remainder. If the remainder is zero, \(x + 1\) is a factor; otherwise, it is not.

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주요 개념

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

Factor Theorem

The Factor Theorem states that a polynomial f(x) has a factor (x - c) if and only if f(c) = 0. To check if a binomial like (x + 1) is a factor, substitute -1 into the polynomial and see if the result is zero.

Synthetic Division

Synthetic division is a shortcut method for dividing a polynomial by a binomial of the form (x - c). It simplifies the division process by using only the coefficients, making it easier to find remainders and verify factors.

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as a product of its factors. Determining if one polynomial divides another without remainder helps break down complex expressions into simpler components.

Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. 2x3+x+2; x+1