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15. -3x(5x + 1)(8 - 2x) 16. (a + b)(3a - b)(2a + 7b) 17. (1 - 2n)^3-7n(…

Question

  1. -3x(5x + 1)(8 - 2x)
  2. (a + b)(3a - b)(2a + 7b)
  3. (1 - 2n)^3-7n(n^2 - 2)
  4. 4(2 - 3w)(w^2 - 2w + 10)
  5. \frac{-42x^{10}y^{5}+12x^{8}y^{3}-6x^{4}y}{6x^{2}y}
  6. \frac{16a^{4}-40a^{2}+24a}{12a^{3}}
  7. the length, width, and height of a rectangular prism can be represented by the expressions (x + 3), (x + 7) and (x - 1). write an expression to represent the surface area of the prism.
  8. the area of a triangle can be represented by the expression 14x^{5}+63x^{3}. if the base is 7x^{3}, write an expression to represent its height.

Explanation:

Response
  1. Problem 15: \(-3x(5x + 1)(8-2x)\)
  • Explanation:
  • Step 1: Multiply \((5x + 1)\) and \((8 - 2x)\)
  • Use the FOIL - method: \((5x+1)(8 - 2x)=5x\times8-5x\times2x + 1\times8-1\times2x=40x-10x^{2}+8 - 2x=-10x^{2}+38x + 8\).
  • Step 2: Multiply by \(-3x\)
  • \(-3x(-10x^{2}+38x + 8)=(-3x)\times(-10x^{2})+(-3x)\times38x+(-3x)\times8 = 30x^{3}-114x^{2}-24x\).
  1. Problem 16: \((a + b)(3a - b)(2a+7b)\)
  • Explanation:
  • Step 1: Multiply \((a + b)\) and \((3a - b)\)
  • Using FOIL - method: \((a + b)(3a - b)=a\times3a-a\times b + b\times3a - b\times b=3a^{2}-ab + 3ab - b^{2}=3a^{2}+2ab - b^{2}\).
  • Step 2: Multiply \((3a^{2}+2ab - b^{2})\) and \((2a + 7b)\)
  • \((3a^{2}+2ab - b^{2})(2a + 7b)=3a^{2}\times2a+3a^{2}\times7b+2ab\times2a+2ab\times7b - b^{2}\times2a - b^{2}\times7b\)
  • \(=6a^{3}+21a^{2}b + 4a^{2}b+14ab^{2}-2ab^{2}-7b^{3}=6a^{3}+25a^{2}b + 12ab^{2}-7b^{3}\).
  1. Problem 17: \((1 - 2n)^{3}-7n(n^{2}-2)\)
  • Explanation:
  • Step 1: Expand \((1 - 2n)^{3}\) using the formula \((a - b)^{3}=a^{3}-3a^{2}b + 3ab^{2}-b^{3}\) where \(a = 1\) and \(b = 2n\)
  • \((1 - 2n)^{3}=1^{3}-3\times1^{2}\times(2n)+3\times1\times(2n)^{2}-(2n)^{3}=1 - 6n + 12n^{2}-8n^{3}\).
  • Step 2: Expand \(-7n(n^{2}-2)\)
  • \(-7n(n^{2}-2)=-7n\times n^{2}+(-7n)\times(-2)=-7n^{3}+14n\).
  • Step 3: Combine like - terms
  • \((1 - 6n + 12n^{2}-8n^{3})+(-7n^{3}+14n)=1 - 6n + 14n+12n^{2}+(-8n^{3}-7n^{3})=1 + 8n+12n^{2}-15n^{3}\).
  1. Problem 18: \(4(2 - 3w)(w^{2}-2w + 10)\)
  • Explanation:
  • Step 1: Multiply \((2 - 3w)\) and \((w^{2}-2w + 10)\)
  • \(2(w^{2}-2w + 10)-3w(w^{2}-2w + 10)=2w^{2}-4w + 20-3w^{3}+6w^{2}-30w\)
  • \(=-3w^{3}+(2w^{2}+6w^{2})+(-4w-30w)+20=-3w^{3}+8w^{2}-34w + 20\).
  • Step 2: Multiply by 4
  • \(4(-3w^{3}+8w^{2}-34w + 20)=-12w^{3}+32w^{2}-136w + 80\).
  1. Problem 19: \(\frac{-42x^{10}y^{5}+12x^{8}y^{3}-6x^{4}y}{6x^{2}y}\)
  • Explanation:
  • Step 1: Divide each term in the numerator by the denominator
  • \(\frac{-42x^{10}y^{5}}{6x^{2}y}+\frac{12x^{8}y^{3}}{6x^{2}y}-\frac{6x^{4}y}{6x^{2}y}\)
  • Using the rule \(\frac{x^{m}}{x^{n}}=x^{m - n}\) and \(\frac{y^{m}}{y^{n}}=y^{m - n}\), we get \(-7x^{10 - 2}y^{5 - 1}+2x^{8 - 2}y^{3 - 1}-x^{4 - 2}y^{1 - 1}=-7x^{8}y^{4}+2x^{6}y^{2}-x^{2}\).
  1. Problem 20: \(\frac{16a^{4}-40a^{2}+24a}{12a^{3}}\)
  • Explanation:
  • Step 1: Divide each term in the numerator by the denominator
  • \(\frac{16a^{4}}{12a^{3}}-\frac{40a^{2}}{12a^{3}}+\frac{24a}{12a^{3}}\)
  • Using the rule \(\frac{a^{m}}{a^{n}}=a^{m - n}\), we get \(\frac{4}{3}a-\frac{10}{3a}+\frac{2}{a^{2}}\).
  1. Problem 21: The length \(l=x + 3\), width \(w=x + 7\), and height \(h=x - 1\) of a rectangular prism. Find the surface - area formula \(S = 2(lw+lh+wh)\)
  • Explanation:
  • Step 1: Calculate \(lw\), \(lh\), and \(wh\)
  • \(lw=(x + 3)(x + 7)=x^{2}+7x+3x + 21=x^{2}+10x + 21\).
  • \(lh=(x + 3)(x - 1)=x^{2}-x+3x - 3=x^{2}+2x - 3\).
  • \(wh=(x + 7)(x - 1)=x^{2}-x+7x - 7=x^{2}+6x - 7\).
  • Step 2: Calculate \(lw+lh+wh\)
  • \((x^{2}+10x + 21)+(x^{2}+2x - 3)+(x^{2}+6x - 7)=3x^{2}+(10x+2x + 6x)+(21 - 3 - 7)=3x^{2}+18x + 11\).
  • Step 3: Calculate \(S\)
  • \(S = 2(3x^{2}+18x + 11)=6x^{2}+36x + 22\).
  1. Problem 22: The area of a triangle \(A = 14x^{5}+63x^{3}\) and base \(b = 7x^{2}\). Use the formula \(A=\frac{1}{2}bh\) to find the height \(h\)
  • Explanation:…

Answer:

  1. \(30x^{3}-114x^{2}-24x\)
  2. \(6a^{3}+25a^{2}b + 12ab^{2}-7b^{3}\)
  3. \(1 + 8n+12n^{2}-15n^{3}\)
  4. \(-12w^{3}+32w^{2}-136w + 80\)
  5. \(-7x^{8}y^{4}+2x^{6}y^{2}-x^{2}\)
  6. \(\frac{4}{3}a-\frac{10}{3a}+\frac{2}{a^{2}}\)
  7. \(6x^{2}+36x + 22\)
  8. \(4x^{3}+18x\)