(x%2B4)(x-5)(x%2B1).jpeg "(x-3)(x+4)(x-5)(x+1)")
Factoring and Finding the Roots of (x-3)(x+4)(x-5)(x+1)
This expression is already in a factored form, which makes it easy to find its roots and understand its behavior.
Understanding the Factored Form
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(x-3)(x+4)(x-5)(x+1) This expression represents the product of four linear factors.
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Linear Factors: Each factor is a simple expression of the form (x - a), where 'a' is a constant. These factors represent lines when graphed.
Finding the Roots
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Roots: The roots of a polynomial are the values of 'x' that make the expression equal to zero. To find the roots, we set each factor equal to zero and solve:
- x - 3 = 0 => x = 3
- x + 4 = 0 => x = -4
- x - 5 = 0 => x = 5
- x + 1 = 0 => x = -1
Therefore, the roots of the expression are x = 3, x = -4, x = 5, and x = -1.
The Graph
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The roots represent the points where the graph of the polynomial intersects the x-axis.
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Since the expression has four factors, the graph will have a degree of 4, meaning it will have a general "W" shape.
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The graph will cross the x-axis at the points x = 3, x = -4, x = 5, and x = -1.
Expanding the Expression (Optional)
While the factored form is useful, we can also expand the expression to see its standard polynomial form:
- Multiply the first two factors: (x-3)(x+4) = x² + x - 12
- Multiply the last two factors: (x-5)(x+1) = x² - 4x - 5
- Multiply the results from step 1 and 2: (x² + x - 12)(x² - 4x - 5) = x⁴ - 3x³ - 21x² + 47x + 60
The expanded form is x⁴ - 3x³ - 21x² + 47x + 60.
In summary, the expression (x-3)(x+4)(x-5)(x+1) is already factored, making it easy to determine its roots (x = 3, x = -4, x = 5, and x = -1) and understand its graphical behavior. By expanding the expression, we can also see its standard polynomial form.