(x%2B2)(x-1)(x-3).jpeg "(x+4)(x+2)(x-1)(x-3)")
Expanding and Simplifying (x+4)(x+2)(x-1)(x-3)
This article will guide you through expanding and simplifying the expression (x+4)(x+2)(x-1)(x-3).
Step 1: Expand the First Two Binomials
We begin by expanding the first two binomials, (x+4)(x+2), using the FOIL method:
- First: x * x = x²
- Outer: x * 2 = 2x
- Inner: 4 * x = 4x
- Last: 4 * 2 = 8
Combining like terms, we get: x² + 2x + 4x + 8 = x² + 6x + 8
Step 2: Expand the Last Two Binomials
Now, we expand the remaining binomials, (x-1)(x-3), using the FOIL method again:
- First: x * x = x²
- Outer: x * -3 = -3x
- Inner: -1 * x = -x
- Last: -1 * -3 = 3
Combining like terms, we obtain: x² - 3x - x + 3 = x² - 4x + 3
Step 3: Multiply the Expanded Binomials
We now have the simplified expressions for the first two binomials and the last two binomials:
- (x+4)(x+2) = x² + 6x + 8
- (x-1)(x-3) = x² - 4x + 3
We multiply these expressions together:
(x² + 6x + 8)(x² - 4x + 3)
To multiply these trinomials, we can use the distributive property:
- x² * (x² - 4x + 3) = x⁴ - 4x³ + 3x²
- 6x * (x² - 4x + 3) = 6x³ - 24x² + 18x
- 8 * (x² - 4x + 3) = 8x² - 32x + 24
Now, we add all these terms together:
x⁴ - 4x³ + 3x² + 6x³ - 24x² + 18x + 8x² - 32x + 24
Step 4: Combine Like Terms
Finally, we combine like terms to get the fully simplified expression:
x⁴ + 2x³ - 13x² - 14x + 24
Therefore, the simplified expression for (x+4)(x+2)(x-1)(x-3) is x⁴ + 2x³ - 13x² - 14x + 24.