Examkin

Soru

Zorluk: ZorPolynomial Factors and Graphs

For a third-degree polynomial p(x)p(x), the expression x2−4x+4x^2 - 4x + 4 is a factor. When p(x)p(x) is divided by x−3x - 3, the remainder is 55, and when p(x)p(x) is divided by x+1x + 1, the remainder is −27-27. What is the remainder when p(x)p(x) is divided by x−1x - 1?

Cevap: 1

Cevap

The remainder when the polynomial is divided by x−1x - 1 is 11.
The correct answer is 11. The polynomial is expressed as p(x)=(x−2)2(ax+b)p(x) = (x-2)^2(ax + b) since it is a third-degree polynomial with a factor of x2−4x+4=(x−2)2x^2 - 4x + 4 = (x-2)^2. Applying the Remainder Theorem, we evaluate the polynomial at x=3x = 3 and x=−1x = -1, giving the system of equations 3a+b=53a + b = 5 and −a+b=−3-a + b = -3. Solving this system yields a=2a = 2 and b=−1b = -1, meaning the polynomial is p(x)=(x−2)2(2x−1)p(x) = (x-2)^2(2x - 1). Finally, the remainder when p(x)p(x) is divided by x−1x - 1 is p(1)=(1−2)2(2(1)−1)=1p(1) = (1-2)^2(2(1) - 1) = 1.

Adım Adım Çözüm

1
Express the third-degree polynomial in terms of its known quadratic factor.
p(x)=(x−2)2(ax+b)p(x) = (x-2)^2(ax + b)
Since the polynomial is of degree 3 and has a quadratic factor of x2−4x+4=(x−2)2x^2 - 4x + 4 = (x-2)^2, the remaining factor must be linear, of the form ax+bax+b.
2
Apply the Remainder Theorem to set up the system of equations.
p(3)=5p(3) = 5 and p(−1)=−27p(-1) = -27
The Remainder Theorem states that the remainder of a polynomial p(x)p(x) when divided by x−cx - c is equal to p(c)p(c).
3
Substitute the values of xx into the polynomial expression.
3a+b=53a + b = 5 and 9(−a+b)=−279(-a + b) = -27
Evaluating p(3)=(3−2)2(3a+b)=5p(3) = (3-2)^2(3a + b) = 5 yields 3a+b=53a + b = 5, and evaluating p(−1)=(−1−2)2(−a+b)=−27p(-1) = (-1-2)^2(-a + b) = -27 yields 9(−a+b)=−279(-a + b) = -27.
4
Simplify the second equation and solve the system of linear equations.
a=2a = 2 and b=−1b = -1
Dividing the second equation by 99 gives −a+b=−3-a + b = -3. Subtracting this from the first equation (3a+b=53a + b = 5) gives 4a=84a = 8, which means a=2a = 2. Substituting a=2a = 2 into the first equation yields b=−1b = -1.
5
Calculate the remainder when p(x)p(x) is divided by x−1x - 1.
p(1)=1p(1) = 1
According to the Remainder Theorem, the remainder of p(x)p(x) divided by x−1x - 1 is p(1)p(1). Substituting x=1x = 1 into p(x)=(x−2)2(2x−1)p(x) = (x-2)^2(2x - 1) yields (1−2)2(2(1)−1)=1(1-2)^2(2(1) - 1) = 1.

Anahtar Kavram

Using the Remainder Theorem and known factors of a polynomial to solve for unknown coefficients.
Bu soruyu puanla

Konular

Polynomial Factors and Graphs

Bu konudan daha fazla soru