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Zorluk: OrtaPolynomial Factors and Graphs

The polynomial p(x)p(x) is defined by p(x)=x3−2x2−kx+6p(x) = x^3 - 2x^2 - kx + 6, where kk is a constant. If x−3x - 3 is a factor of p(x)p(x), which of the following is a factor of p(x)−2x+8p(x) - 2x + 8?

  1. A
    x−1x - 1
  2. x−2x - 2Cevap
  3. C
    x+2x + 2
  4. D
    x+3x + 3

Cevap

The expression x−2x - 2 is a factor of the modified polynomial.
The correct answer is the binomial expression x−2x - 2. By the Factor Theorem, x−3x - 3 being a factor of p(x)p(x) implies p(3)=0p(3) = 0. Substituting x=3x = 3 into p(x)=x3−2x2−kx+6p(x) = x^3 - 2x^2 - kx + 6 gives 27−18−3k+6=027 - 18 - 3k + 6 = 0, which simplifies to 15−3k=015 - 3k = 0, yielding k=5k = 5. Thus, p(x)=x3−2x2−5x+6p(x) = x^3 - 2x^2 - 5x + 6. The modified expression is p(x)−2x+8=x3−2x2−7x+14p(x) - 2x + 8 = x^3 - 2x^2 - 7x + 14. Factoring this by grouping gives x2(x−2)−7(x−2)=(x2−7)(x−2)x^2(x - 2) - 7(x - 2) = (x^2 - 7)(x - 2), showing that x−2x - 2 is a factor.

Adım Adım Çözüm

1
Apply the Factor Theorem to find the value of kk. Since x−3x - 3 is a factor of p(x)p(x), we must have p(3)=0p(3) = 0.
Substituting x=3x = 3 into the equation gives 33−2(3)2−k(3)+6=03^3 - 2(3)^2 - k(3) + 6 = 0, which simplifies to 27−18−3k+6=027 - 18 - 3k + 6 = 0, or 15−3k=015 - 3k = 0. Solving for kk yields k=5k = 5.
This allows us to determine the complete polynomial expression for p(x)p(x).
2
Substitute the value of kk back into p(x)p(x) and determine the expression for the modified polynomial, q(x)=p(x)−2x+8q(x) = p(x) - 2x + 8.
Substituting k=5k = 5 gives p(x)=x3−2x2−5x+6p(x) = x^3 - 2x^2 - 5x + 6. Therefore, the modified polynomial is q(x)=(x3−2x2−5x+6)−2x+8=x3−2x2−7x+14q(x) = (x^3 - 2x^2 - 5x + 6) - 2x + 8 = x^3 - 2x^2 - 7x + 14.
This simplifies the expression so we can find its factors.
3
Factor the modified polynomial q(x)=x3−2x2−7x+14q(x) = x^3 - 2x^2 - 7x + 14 by grouping.
Grouping the terms gives x2(x−2)−7(x−2)x^2(x - 2) - 7(x - 2). Factoring out the common binomial term (x−2)(x - 2) results in (x2−7)(x−2)(x^2 - 7)(x - 2).
This reveals the individual linear and quadratic factors of the polynomial.
4
Identify which of the options is a factor of the modified polynomial.
The factors of the polynomial are x−2x - 2 and x2−7x^2 - 7. Thus, x−2x - 2 is a factor.
This answers the question by matching our factored form with the options.

Anahtar Kavram

The Factor Theorem states that a linear expression x−cx - c is a factor of a polynomial f(x)f(x) if and only if f(c)=0f(c) = 0. We can also factor cubic polynomials by grouping terms to find their roots and factors.

Alternatif Yöntem

Alternatively, instead of fully factoring the polynomial, one can apply the Factor Theorem directly to the choices. Since a linear expression x−cx - c is a factor of q(x)=p(x)−2x+8q(x) = p(x) - 2x + 8 if and only if q(c)=0q(c) = 0, we can substitute the root cc corresponding to each choice into q(x)q(x) after finding k=5k = 5. For the correct factor x−2x - 2, substituting x=2x = 2 yields q(2)=23−2(2)2−7(2)+14=8−8−14+14=0q(2) = 2^3 - 2(2)^2 - 7(2) + 14 = 8 - 8 - 14 + 14 = 0.
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