(x-2)(x%2B4)(3x%2B7).jpeg "(x+1)(x-2)(x+4)(3x+7)")
Expanding the Expression (x+1)(x-2)(x+4)(3x+7)
This article will guide you through the process of expanding the given expression: (x+1)(x-2)(x+4)(3x+7). This involves multiplying the factors together in a systematic way.
Step 1: Expand the first two factors
Let's start by expanding the first two factors: (x+1)(x-2). We can use the FOIL method (First, Outer, Inner, Last) or the distributive property:
- FOIL method:
- First: x * x = x²
- Outer: x * -2 = -2x
- Inner: 1 * x = x
- Last: 1 * -2 = -2
- Distributive Property: (x + 1)(x - 2) = x(x - 2) + 1(x - 2) = x² - 2x + x - 2
Combining like terms, we get: x² - x - 2
Step 2: Expand the remaining two factors
Now, let's expand the remaining two factors: (x+4)(3x+7). We can use the same methods as before:
- FOIL method:
- First: x * 3x = 3x²
- Outer: x * 7 = 7x
- Inner: 4 * 3x = 12x
- Last: 4 * 7 = 28
- Distributive Property: (x + 4)(3x + 7) = x(3x + 7) + 4(3x + 7) = 3x² + 7x + 12x + 28
Combining like terms, we get: 3x² + 19x + 28
Step 3: Multiply the expanded results
We are now left with two simplified expressions: x² - x - 2 and 3x² + 19x + 28. We need to multiply these together:
(x² - x - 2)(3x² + 19x + 28)
Again, we can use the distributive property or a more systematic way of multiplying each term of the first expression by each term of the second expression:
- x² * (3x² + 19x + 28) = 3x⁴ + 19x³ + 28x²
- -x * (3x² + 19x + 28) = -3x³ - 19x² - 28x
- -2 * (3x² + 19x + 28) = -6x² - 38x - 56
Finally, combine all the terms: 3x⁴ + 16x³ + 3x² - 66x - 56
Conclusion
The expanded form of the expression (x+1)(x-2)(x+4)(3x+7) is 3x⁴ + 16x³ + 3x² - 66x - 56. This process involves applying the distributive property or FOIL method multiple times to simplify the expression.