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The polynomial p(x)=3x3−20x2+37x−20p(x)=3x^3-20x^2+37x-20 has a known factor of (x−4)(x-4). Rewrite p(x)p(x) as a product of linear factors. p(x)=p(x)=

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Q. The polynomial p(x)=3x3−20x2+37x−20p(x)=3x^3-20x^2+37x-20 has a known factor of (x−4)(x-4). Rewrite p(x)p(x) as a product of linear factors. p(x)=p(x)=
  1. Divide by (x−4-4): Use polynomial division to divide p(x)p(x) by (x−4)(x-4).\newlinep(x)=3x3−20x2+37x−20p(x) = 3x^3 - 20x^2 + 37x - 20\newlineDivide by (x−4)(x-4):\newline3x3−20x2+37x−20÷(x−4)3x^3 - 20x^2 + 37x - 20 \div (x-4)
  2. Perform the division: Perform the division.\newlineFirst term: 3x3x=3x2\frac{3x^3}{x} = 3x^2\newlineMultiply: 3x2⋅(x−4)=3x3−12x23x^2 \cdot (x-4) = 3x^3 - 12x^2\newlineSubtract: (3x3−20x2)−(3x3−12x2)=−8x2(3x^3 - 20x^2) - (3x^3 - 12x^2) = -8x^2
  3. Continue the division: Continue the division.\newlineNext term: −8x2x=−8x\frac{-8x^2}{x} = -8x\newlineMultiply: −8x⋅(x−4)=−8x2+32x-8x \cdot (x-4) = -8x^2 + 32x\newlineSubtract: (−8x2+37x)−(−8x2+32x)=5x(-8x^2 + 37x) - (-8x^2 + 32x) = 5x
  4. Continue the division: Continue the division.\newlineNext term: 5xx=5\frac{5x}{x} = 5\newlineMultiply: 5⋅(x−4)=5x−205 \cdot (x-4) = 5x - 20\newlineSubtract: (5x−20)−(5x−20)=0(5x - 20) - (5x - 20) = 0
  5. Write the quotient: Write the quotient.\newlineQuotient: 3x2−8x+53x^2 - 8x + 5\newlineSo, p(x)=(x−4)(3x2−8x+5)p(x) = (x-4)(3x^2 - 8x + 5)
  6. Factor the quadratic: Factor the quadratic 3x2−8x+53x^2 - 8x + 5.\newlineFind factors of 3⋅5=153 \cdot 5 = 15 that add to −8-8: −3-3 and −5-5\newlineRewrite: 3x2−3x−5x+53x^2 - 3x - 5x + 5\newlineFactor by grouping: 3x(x−1)−5(x−1)3x(x-1) - 5(x-1)\newlineFactor out common term: (x−1)(3x−5)(x-1)(3x-5)
  7. Write p(x) as a product: Write p(x)p(x) as a product of linear factors.\newlinep(x)=(x−4)(x−1)(3x−5)p(x) = (x-4)(x-1)(3x-5)

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