(x-2)(x%2B4)(3x%2B7)-in-standard-form.jpeg "(x+1)(x-2)(x+4)(3x+7) In Standard Form")
Expanding and Simplifying (x+1)(x-2)(x+4)(3x+7)
This problem involves expanding and simplifying a polynomial expression. Let's break down the steps:
1. Expanding the First Two Factors:
Start by expanding the first two factors: (x+1)(x-2)
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Use the FOIL method (First, Outer, Inner, Last) or the distributive property:
(x + 1)(x - 2) = x² - 2x + x - 2 = x² - x - 2
2. Expanding the Last Two Factors:
Next, expand the last two factors: (x+4)(3x+7)
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Again, use FOIL or the distributive property:
(x + 4)(3x + 7) = 3x² + 7x + 12x + 28 = 3x² + 19x + 28
3. Expanding the Two Resulting Binomials:
Now we have: (x² - x - 2) (3x² + 19x + 28)
- We need to multiply these two binomials. This can be done systematically:
- Multiply x² from the first binomial by each term in the second binomial: x² * (3x² + 19x + 28) = 3x⁴ + 19x³ + 28x²
- Multiply -x from the first binomial by each term in the second binomial: -x * (3x² + 19x + 28) = -3x³ - 19x² - 28x
- Multiply -2 from the first binomial by each term in the second binomial: -2 * (3x² + 19x + 28) = -6x² - 38x - 56
4. Combining Like Terms:
Finally, combine the like terms from the expanded products:
3x⁴ + 19x³ + 28x² - 3x³ - 19x² - 28x - 6x² - 38x - 56
= 3x⁴ + 16x³ + 3x² - 66x - 56
The Standard Form:
Therefore, the polynomial (x+1)(x-2)(x+4)(3x+7) in standard form is 3x⁴ + 16x³ + 3x² - 66x - 56.