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The polynomial \newlinep(x)=5x3−44x2+61x+14p(x)=5x^{3}-44x^{2}+61x+14 has a known factor of \newline(x−7)(x-7).\newlineRewrite \newlinep(x)p(x) as a product of linear factors.\newlinep(x)=p(x)=

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Q. The polynomial \newlinep(x)=5x3−44x2+61x+14p(x)=5x^{3}-44x^{2}+61x+14 has a known factor of \newline(x−7)(x-7).\newlineRewrite \newlinep(x)p(x) as a product of linear factors.\newlinep(x)=p(x)=
  1. Perform Polynomial Division: Step 11: Perform polynomial division to divide p(x)p(x) by (x−7)(x-7).\newlineCalculation: Using synthetic division with root 77,\newline5x3−44x2+61x+145x^3 - 44x^2 + 61x + 14 divided by x−7x - 7:\newline\begin{array}{r|rrrr} 7 & 5 & -44 & 61 & 14 \ \hline & & 35 & -63 & -14 \ \hline & 5 & -9 & -2 & 0 \end{array}\newlineResult: 5x2−9x−25x^2 - 9x - 2
  2. Factorize Quadratic Equation: Step 22: Factorize the quadratic equation 5x2−9x−25x^2 - 9x - 2.\newlineCalculation: Using the AC method, where A=5A=5, B=−9B=-9, C=−2C=-2,\newlineProduct AC=−10AC = -10, Sum B=−9B = -9,\newlineFactors of −10-10 that add up to −9-9 are −10-10 and 11,\newlineRewrite middle term: A=5A=500,\newlineGroup: A=5A=511,\newlineFactor out common terms: A=5A=522,\newlineFactor by grouping: A=5A=533
  3. Write as Linear Factors: Step 33: Write p(x)p(x) as a product of linear factors.\newlineCalculation: p(x)=(x−7)(5x+1)(x−2)p(x) = (x - 7)(5x + 1)(x - 2)

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