Answer :
Sure, let's break down the given polynomial \( P_3(x) \) step by step and then expand it fully to arrive at the simplified form.
The polynomial is given by:
[tex]\[ P_3(x) = 1 + (x+1) - \frac{2}{3}(x+1)(x-1) - \frac{1}{12}(x+1)(x-1)(x-2) \][/tex]
### Step-by-Step Expansion
1. Simplify each term separately:
- The first term is simply \( 1 \).
- The second term is \( (x + 1) \).
- The third term is \( - \frac{2}{3}(x+1)(x-1) \):
[tex]\[ (x + 1)(x - 1) = x^2 - 1 \][/tex]
Therefore:
[tex]\[ - \frac{2}{3}(x^2 - 1) = - \frac{2}{3}x^2 + \frac{2}{3} \][/tex]
- The fourth term is \( - \frac{1}{12}(x+1)(x-1)(x-2) \):
[tex]\[ (x + 1)(x - 1)(x - 2) = (x^2 - 1)(x - 2) = x^3 - 2x^2 - x + 2 \][/tex]
Therefore:
[tex]\[ -\frac{1}{12}(x^3 - 2x^2 - x + 2) = - \frac{1}{12}x^3 + \frac{1}{6}x^2 + \frac{1}{12}x - \frac{1}{6} \][/tex]
2. Combine all the terms together:
Summarize all the contributions from each term into \( P_3(x) \):
[tex]\[ P_3(x) = 1 + (x + 1) - \frac{2}{3}(x^2 - 1) - \frac{1}{12}(x^3 - 2x^2 - x + 2) \][/tex]
Substitute and combine:
[tex]\[ P_3(x) = 1 + x + 1 - \frac{2}{3}x^2 + \frac{2}{3} - \frac{1}{12}x^3 + \frac{1}{6}x^2 + \frac{1}{12}x - \frac{1}{6} \][/tex]
3. Combine all like terms:
- Constants:
[tex]\[ 1 + 1 + \frac{2}{3} - \frac{1}{6} = \frac{3}{3} + \frac{2}{3} - \frac{1}{6} = 2 + \frac{1}{6} = 2.5 \][/tex]
- \( x \) terms:
[tex]\[ x + \frac{1}{12}x = 1.08333333333333 x \][/tex]
- \( x^2 \) terms:
[tex]\[ -\frac{2}{3}x^2 + \frac{1}{6}x^2 = -\frac{4}{6}x^2 + \frac{1}{6}x^2 = - \frac{1}{2}x^2 \][/tex]
- \( x^3 \) terms:
[tex]\[ - \frac{1}{12}x^3 = -0.0833333333333333 x^3 \][/tex]
Final combined polynomial:
[tex]\[ P_3(x) = -0.0833333333333333 x^3 - 0.5 x^2 + 1.08333333333333 x + 2.5 \][/tex]
Therefore,
[tex]\[ P_3(x) = 1 + (x+1) - \frac{2}{3}(x+1)(x-1) - \frac{1}{12}(x+1)(x-1)(x-2) \][/tex]
expands and simplifies to:
[tex]\[ P_3(x) = -0.0833333333333333 x^3 - 0.5 x^2 + 1.08333333333333 x + 2.5 \][/tex]
The polynomial is given by:
[tex]\[ P_3(x) = 1 + (x+1) - \frac{2}{3}(x+1)(x-1) - \frac{1}{12}(x+1)(x-1)(x-2) \][/tex]
### Step-by-Step Expansion
1. Simplify each term separately:
- The first term is simply \( 1 \).
- The second term is \( (x + 1) \).
- The third term is \( - \frac{2}{3}(x+1)(x-1) \):
[tex]\[ (x + 1)(x - 1) = x^2 - 1 \][/tex]
Therefore:
[tex]\[ - \frac{2}{3}(x^2 - 1) = - \frac{2}{3}x^2 + \frac{2}{3} \][/tex]
- The fourth term is \( - \frac{1}{12}(x+1)(x-1)(x-2) \):
[tex]\[ (x + 1)(x - 1)(x - 2) = (x^2 - 1)(x - 2) = x^3 - 2x^2 - x + 2 \][/tex]
Therefore:
[tex]\[ -\frac{1}{12}(x^3 - 2x^2 - x + 2) = - \frac{1}{12}x^3 + \frac{1}{6}x^2 + \frac{1}{12}x - \frac{1}{6} \][/tex]
2. Combine all the terms together:
Summarize all the contributions from each term into \( P_3(x) \):
[tex]\[ P_3(x) = 1 + (x + 1) - \frac{2}{3}(x^2 - 1) - \frac{1}{12}(x^3 - 2x^2 - x + 2) \][/tex]
Substitute and combine:
[tex]\[ P_3(x) = 1 + x + 1 - \frac{2}{3}x^2 + \frac{2}{3} - \frac{1}{12}x^3 + \frac{1}{6}x^2 + \frac{1}{12}x - \frac{1}{6} \][/tex]
3. Combine all like terms:
- Constants:
[tex]\[ 1 + 1 + \frac{2}{3} - \frac{1}{6} = \frac{3}{3} + \frac{2}{3} - \frac{1}{6} = 2 + \frac{1}{6} = 2.5 \][/tex]
- \( x \) terms:
[tex]\[ x + \frac{1}{12}x = 1.08333333333333 x \][/tex]
- \( x^2 \) terms:
[tex]\[ -\frac{2}{3}x^2 + \frac{1}{6}x^2 = -\frac{4}{6}x^2 + \frac{1}{6}x^2 = - \frac{1}{2}x^2 \][/tex]
- \( x^3 \) terms:
[tex]\[ - \frac{1}{12}x^3 = -0.0833333333333333 x^3 \][/tex]
Final combined polynomial:
[tex]\[ P_3(x) = -0.0833333333333333 x^3 - 0.5 x^2 + 1.08333333333333 x + 2.5 \][/tex]
Therefore,
[tex]\[ P_3(x) = 1 + (x+1) - \frac{2}{3}(x+1)(x-1) - \frac{1}{12}(x+1)(x-1)(x-2) \][/tex]
expands and simplifies to:
[tex]\[ P_3(x) = -0.0833333333333333 x^3 - 0.5 x^2 + 1.08333333333333 x + 2.5 \][/tex]
