(x-2)(x%2B6).jpeg "(x+1)(x-2)(x+6)")
Expanding and Factoring the Expression (x+1)(x-2)(x+6)
This article explores the process of expanding and factoring the expression (x+1)(x-2)(x+6).
Expanding the Expression
To expand the expression, we use the distributive property. This involves multiplying each term in the first factor by every term in the second factor, and then repeating this process with the third factor.
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First Step: Expand (x+1)(x-2)
- (x+1)(x-2) = x(x-2) + 1(x-2)
- = x² - 2x + x - 2
- = x² - x - 2
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Second Step: Multiply the result by (x+6)
- (x² - x - 2)(x+6) = x²(x+6) - x(x+6) - 2(x+6)
- = x³ + 6x² - x² - 6x - 2x - 12
- = x³ + 5x² - 8x - 12
Therefore, the expanded form of (x+1)(x-2)(x+6) is x³ + 5x² - 8x - 12.
Factoring the Expression
Factoring the expression involves finding the factors that, when multiplied together, equal the original expression. In this case, we already know the factors: (x+1), (x-2), and (x+6).
However, let's assume we only have the expanded form x³ + 5x² - 8x - 12. To factor this expression, we would need to follow these steps:
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Find a possible root: We can use the Rational Root Theorem to find possible rational roots. The theorem states that any rational root of a polynomial must be of the form p/q, where p is a factor of the constant term (-12) and q is a factor of the leading coefficient (1).
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Test the possible roots: We can use synthetic division or direct substitution to test each possible root until we find one that works.
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Factor the resulting quadratic: Once a root is found, we can factor the resulting quadratic expression.
In this case, the factoring process would lead us back to the original factored form: (x+1)(x-2)(x+6).
Applications
Understanding how to expand and factor expressions like (x+1)(x-2)(x+6) is crucial in various mathematical fields, including:
- Algebra: Solving equations and inequalities.
- Calculus: Finding derivatives and integrals.
- Discrete mathematics: Combinatorics and graph theory.
- Physics: Modeling physical phenomena.
By mastering these techniques, we gain a deeper understanding of algebraic expressions and their applications in different areas of mathematics and beyond.