Factor completely: 3f² + 8f - 3
Understand the Problem
The question is asking to factor the expression 3f² + 8f - 3 completely.
Answer
The expression factors to $(3f - 1)(f + 3)$.
Answer for screen readers
The completely factored form of the expression is $(3f - 1)(f + 3)$.
Steps to Solve
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Identify coefficients and terms The expression is $3f^2 + 8f - 3$. Here, (a = 3), (b = 8), and (c = -3).
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Calculate the product of (a) and (c) Calculate (ac): $$ ac = 3 \cdot (-3) = -9 $$
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Find factors of (ac) that sum to (b) We need two numbers that multiply to (-9) and add to (8). The numbers are (9) and (-1) because: $$ 9 \cdot (-1) = -9 $$ $$ 9 + (-1) = 8 $$
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Rewrite the middle term using the found factors The expression can be rewritten as: $$ 3f^2 + 9f - 1f - 3 $$
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Group the terms Group the first two terms and the last two terms: $$ (3f^2 + 9f) + (-1f - 3) $$
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Factor by grouping Factor out the common factors in each group: $$ 3f(f + 3) - 1(f + 3) $$
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Factor out the common binomial factor Combine the factors: $$ (3f - 1)(f + 3) $$
The completely factored form of the expression is $(3f - 1)(f + 3)$.
More Information
Factoring quadratic expressions involves finding two numbers that multiply to (ac) (product of the leading coefficient and the constant term) and add to (b) (the coefficient of the middle term). This method helps to break down the polynomial into simpler factors.
Tips
- Forgetting to check if the factors found indeed add up to (b).
- Misplacing signs when factoring, especially with negative constants.
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