Welcome to binomial multiplication! Today we'll solve the expression (x-2)(x+5). A binomial is a polynomial with exactly two terms. When we multiply two binomials, we need to multiply every term in the first binomial by every term in the second binomial. Let's see how this works step by step.
Now let's learn the FOIL method, a systematic way to multiply binomials. FOIL stands for First, Outer, Inner, and Last. Using our expression (x-2)(x+5), we multiply: First terms x times x, Outer terms x times 5, Inner terms negative 2 times x, and Last terms negative 2 times 5. The arrows show which terms we're multiplying at each step.
Now let's perform each FOIL calculation step by step. First, we multiply the first terms: x times x equals x squared. Next, the outer terms: x times 5 equals 5x. Then the inner terms: negative 2 times x equals negative 2x. Finally, the last terms: negative 2 times 5 equals negative 10. Combining all these results, we get x squared plus 5x minus 2x minus 10.
Now we need to combine like terms in our expanded expression x squared plus 5x minus 2x minus 10. The like terms are 5x and negative 2x, which are both x terms. We combine them: 5x minus 2x equals 3x. This gives us our final simplified result: x squared plus 3x minus 10.
Let's verify our answer using an alternative method: the distributive property. We can rewrite (x-2)(x+5) as x(x+5) minus 2(x+5). Distributing gives us x squared plus 5x minus 2x minus 10, which simplifies to x squared plus 3x minus 10. This matches our FOIL result exactly! Both methods confirm that our answer is correct.