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The polynomial p(x)=x^(3)-6x^(2)+32 has a known factor of (x-4). Rewrite p(x) as a product of linear factors. p(x)=

The polynomial p(x)=x3−6x2+32 p(x)=x^{3}-6 x^{2}+32 has a known factor of (x−4) (x-4) .\newlineRewrite p(x) p(x) as a product of linear factors.\newlinep(x)= p(x)=

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Q. The polynomial p(x)=x3−6x2+32 p(x)=x^{3}-6 x^{2}+32 has a known factor of (x−4) (x-4) .\newlineRewrite p(x) p(x) as a product of linear factors.\newlinep(x)= p(x)=
  1. Perform polynomial division: Given that (x−4)(x-4) is a known factor of p(x)p(x), we will perform polynomial division to divide p(x)p(x) by (x−4)(x-4). Reasoning: To find the other factors, we need to divide the polynomial by the known factor. Calculation: Perform the division of p(x)p(x) by (x−4)(x-4). Math error check:
  2. Set up the division: Set up the division of p(x)p(x) by (x−4)(x-4).\newlineReasoning: We will use either long division or synthetic division to divide the polynomial.\newlineCalculation: \newlinep(x)=x3−6x2+0x+32p(x) = x^3 - 6x^2 + 0x + 32 (Note: The term 0x0x is added for the missing x-term)\newlineDivide by (x−4)(x-4).\newlineMath error check:
  3. Perform the division: Perform the division.\newlineReasoning: By dividing, we will find the quotient which will give us the other factors of p(x)p(x).\newlineCalculation: \newlineWhen we divide x3−6x2+0x+32x^3 - 6x^2 + 0x + 32 by (x−4)(x-4), we get x2−2x−8x^2 - 2x - 8 as the quotient.\newlineMath error check:
  4. Factor the quotient: Factor the quotient x2−2x−8x^2 - 2x - 8. Reasoning: The quotient is a quadratic polynomial, which can be factored into linear factors if it has real roots. Calculation: x2−2x−8=(x−4)(x+2)x^2 - 2x - 8 = (x - 4)(x + 2) Math error check:
  5. Write p(x)p(x) as a product: Write p(x)p(x) as a product of linear factors.\newlineReasoning: Since we have factored the quotient and we know (x−4)(x-4) is a factor, we can write p(x)p(x) as the product of these factors.\newlineCalculation: \newlinep(x)=(x−4)(x−4)(x+2)p(x) = (x - 4)(x - 4)(x + 2)\newlineMath error check:

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