교과서 질문
Graph each function. Determine the largest open intervals of the domain over which each function is (a) increasing or (b) decreasing. ƒ(x)=(1/2)(x-2)2+4
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Ch. 3 - Polynomial and Rational Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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4장, 문제 19
Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first.
검증된 단계별 안내1
Identify the divisor polynomial and rewrite it in the form \(x - c\). Since the divisor is \(x + 3\), rewrite it as \(x - (-3)\), so \(c = -3\).
Apply the Factor Theorem by evaluating the first polynomial at \(x = -3\). This means substituting \(-3\) into \(2x^4 + 5x^3 - 2x^2 + 5x + 6\) and calculating the result.
If the result from step 2 is zero, then \(x + 3\) is a factor of the polynomial. If not, it is not a factor.
To confirm, perform synthetic division of the first polynomial by \(x + 3\) using \(c = -3\). Set up the synthetic division with the coefficients of the polynomial: 2, 5, -2, 5, 6.
Carry out the synthetic division step-by-step: bring down the first coefficient, multiply by \(c\), add to the next coefficient, and repeat until all coefficients are processed. The remainder will be the last value obtained. If the remainder is zero, \(x + 3\) is a factor; otherwise, it is not.
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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
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3m주요 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 + 3 is a factor, substitute -3 into the polynomial and see if the result is zero. If it is, then x + 3 divides the polynomial exactly.
추천 영상:
04:36
Factor by Grouping
Synthetic Division
Synthetic division is a shortcut method for dividing a polynomial by a linear binomial of the form x - c. It simplifies the long division process by using only the coefficients, making it faster to find the quotient and remainder. If the remainder is zero, the divisor is a factor.
추천 영상:
05:10Higher Powers of i
Polynomial Factorization
Polynomial factorization involves expressing a polynomial as a product of its factors. Identifying factors helps simplify expressions and solve polynomial equations. Using the Factor Theorem and synthetic division together aids in breaking down complex polynomials into simpler components.
추천 영상:
07:30Introduction to Factoring Polynomials
관련 실천
교과서 질문
Solve each quadratic inequality. Give the solution set in interval notation. x2 - 2 > x
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교과서 질문
Use an end behavior diagram, as shown below, to describe the end behavior of the graph of each polynomial function. ƒ(x)=5x5+2x3-3x+4
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교과서 질문
Match each statement with its corresponding graph in choices A–D. In each case, k > 0. y varies directly as the second power of x. (y=kx2)
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교과서 질문
Graph the following on the same coordinate system.
(a) y = x2
(b) y = 3x2
(c) y = 1/3x2
(d) How does the coefficient of x2 affect the shape of the graph?
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교과서 질문
Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1.
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