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#6. Factor completely.

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Type your answers to each problem in the table below. The order of the factors doesn't matter, but use parentheses around each one!
Ex) 2x ^ 2 + 5x - 3
6) 2x ^ 2 + x - 10
Example
(2x-1)
Correct?
(x + 3)

1 answer

To factor the expression \(2x^2 + x - 10\) completely, we can follow these steps:

  1. Look for two numbers that multiply to \(a \cdot c\) (where \(a\) is the coefficient of \(x^2\) and \(c\) is the constant term) and add to \(b\) (the coefficient of \(x\)). Here, \(a = 2\), \(b = 1\), and \(c = -10\). So, we need two numbers that multiply to \(2 \cdot (-10) = -20\) and add up to \(1\).

  2. The two numbers that meet these criteria are 5 and -4 since \(5 \times (-4) = -20\) and \(5 + (-4) = 1\).

  3. Rewrite the middle term using these numbers: \[ 2x^2 + 5x - 4x - 10 \]

  4. Group the terms: \[ (2x^2 + 5x) + (-4x - 10) \]

  5. Factor out common factors from each group: \[ x(2x + 5) - 2(2x + 5) \]

  6. Factor out the common binomial factor \((2x + 5)\): \[ (2x + 5)(x - 2) \]

So, the complete factorization of \(2x^2 + x - 10\) is: \[ (2x + 5)(x - 2) \]

You can enter the answer as: (2x + 5)(x - 2)