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Find the value of c so that the polynomial p(x) is divisible by (x-3). p(x)=-x^(3)+cx^(2)-4x+3 c=

Find the value of c c so that the polynomial p(x) p(x) is divisible by (x−3) (x-3) .\newlinep(x)=−x3+cx2−4x+3 p(x)=-x^{3}+c x^{2}-4 x+3 \newlinec= c=

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Composition of linear and quadratic functions: find a valueright chevron icon

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Q. Find the value of c c so that the polynomial p(x) p(x) is divisible by (x−3) (x-3) .\newlinep(x)=−x3+cx2−4x+3 p(x)=-x^{3}+c x^{2}-4 x+3 \newlinec= c=
  1. Use Remainder Theorem: To determine the value of cc that makes the polynomial p(x)p(x) divisible by (x−3)(x-3), we need to use the Remainder Theorem. According to the Remainder Theorem, if a polynomial p(x)p(x) is divisible by (x−a)(x-a), then p(a)=0p(a) = 0. In this case, aa is 33, so we need to find the value of cc such that p(3)=0p(3) = 0.
  2. Substitute x=3x = 3: Let's substitute x=3x = 3 into the polynomial p(x)=−x3+cx2−4x+3p(x) = -x^3 + cx^2 - 4x + 3 and set it equal to zero.\newlinep(3)=−(3)3+c(3)2−4(3)+3=0p(3) = -(3)^3 + c(3)^2 - 4(3) + 3 = 0
  3. Calculate p(3)p(3): Now, we calculate the value of p(3)p(3) with the given values.p(3)=−27+9c−12+3p(3) = -27 + 9c - 12 + 3
  4. Simplify the expression: Simplify the expression to find the value of cc.−27+9c−12+3=0-27 + 9c - 12 + 3 = 09c−36=09c - 36 = 0
  5. Solve for c: Solve for c by adding 3636 to both sides of the equation.\newline9c=369c = 36
  6. Solve for c: Solve for c by adding 3636 to both sides of the equation.\newline9c=369c = 36 Divide both sides by 99 to isolate cc.\newlinec=369c = \frac{36}{9}\newlinec=4c = 4

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