Problem 26 - Introductory Algebra for College Students

Question

Factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$x^{2}+4 x-10$$

Step-by-Step Solution

Answer

The factored form of the trinomial \(x^{2}+4 x-10\) is \((x-2)(x+5)\).

Step 1: Identify the Variables in the Trinomial

The trinomial being discussed in the problem is \(x^{2}+4x-10\). Here, we can identify that \(a=1\), \(b=4\) and \(c=-10\) as components of the standard form quadratic trinomial \(ax^{2}+bx+c\).

Step 2: Factor the Trinomial

To factor the trinomial, we need to find two numbers that when multiplied, give \(-10\) (the product of 'a' and 'c') and when added, give \(4\) (the value of 'b'). The numbers which satisfy this condition are \(-2\) and \(5\). Therefore, we can write our trinomial as: \(x^{2} - 2x + 5x - 10\). We can now group the terms and factor by grouping, giving us: \((x^2 - 2x) + (5x - 10)\). This further simplifies to: \(x(x-2) + 5(x-2)\). Finally, we factor out the common binomial factor to give the factored form of the trinomial: \((x-2)(x+5)\).

Step 3: Check Factorization Using FOIL

Now we will check our factored form using the FOIL method. To do this, we multiply the 'first' terms, then the 'outer' terms, then the 'inner' terms and finally the 'last' terms and add all of these together. Doing this, we get: \(x * x + x * 5 + (-2) * x + (-2) * 5 = x^{2} + 3x - 10\). This is the original trinomial, hence our answer is correct.