(x%2B1)(x-4).jpeg "(x+3)(x+1)(x-4)")
Factoring and Expanding (x+3)(x+1)(x-4)
This expression represents the product of three binomials: (x+3), (x+1), and (x-4). We can explore its properties by expanding and factoring it.
Expanding the Expression
To expand the expression, we can use the distributive property (sometimes called FOIL) multiple times.
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Start with the first two binomials: (x+3)(x+1) = x² + x + 3x + 3 = x² + 4x + 3
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Multiply the result by the third binomial: (x² + 4x + 3)(x-4) = x³ - 4x² + 4x² - 16x + 3x - 12 = x³ - 13x - 12
Therefore, the expanded form of (x+3)(x+1)(x-4) is x³ - 13x - 12.
Factoring the Expression
We can also factor the expression by reversing the expansion process. Here's how:
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Identify the roots: The roots of the expression are the values of x that make the expression equal to zero. We can find the roots by setting the expression equal to zero and solving for x: x³ - 13x - 12 = 0 This can be factored as: (x+3)(x+1)(x-4) = 0 Therefore, the roots are x = -3, x = -1, and x = 4.
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Construct the factored form: Since we know the roots, we can write the factored form as: (x+3)(x+1)(x-4)
Conclusion
The expression (x+3)(x+1)(x-4) represents a cubic polynomial. By expanding the expression, we get x³ - 13x - 12. By factoring the expression, we can identify its roots and express it as (x+3)(x+1)(x-4). Understanding how to expand and factor expressions like this is crucial in algebra, as it allows us to manipulate and solve equations.