(x%2B1)(x%2B5)%2B(x%2B3)%5E3.jpeg "(3x+2)(x+1)(x+5)+(x+3)^3")
Expanding and Simplifying the Expression (3x+2)(x+1)(x+5)+(x+3)^3
This article will guide you through the process of expanding and simplifying the algebraic expression: (3x+2)(x+1)(x+5)+(x+3)^3.
Step 1: Expanding the First Part of the Expression
We begin by expanding the product of the first three binomials: (3x+2)(x+1)(x+5).
a) Multiplying the first two binomials: (3x+2)(x+1) = 3x² + 3x + 2x + 2 = 3x² + 5x + 2
b) Multiplying the result by the third binomial: (3x² + 5x + 2)(x+5) = 3x³ + 15x² + 5x² + 25x + 2x + 10 = 3x³ + 20x² + 27x + 10
Step 2: Expanding the Second Part of the Expression
Next, we expand the cube of the binomial: (x+3)^3
a) Applying the binomial theorem: (x+3)^3 = x³ + 3(x²)(3) + 3(x)(3²) + 3³ = x³ + 9x² + 27x + 27
Step 3: Combining the Expanded Parts
Now we have: 3x³ + 20x² + 27x + 10 + x³ + 9x² + 27x + 27
Step 4: Simplifying the Expression
Finally, we combine like terms: (3x³ + x³) + (20x² + 9x²) + (27x + 27x) + (10 + 27)
This gives us the simplified expression: 4x³ + 29x² + 54x + 37
Therefore, the expanded and simplified form of the expression (3x+2)(x+1)(x+5)+(x+3)^3 is 4x³ + 29x² + 54x + 37.