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Chapter 0: Problem 19

Factor each trinomial, or state that the trinomial is prime. $$x^{2}-2 x-15$$

Short Answer

Expert verified
The factored form of the trinomial \(x^{2}-2x -15\) is \((x - 5)(x + 3)\).

Step by step solution

01

Understand the trinomial pattern

Confirm that the given trinomial is in the standard form \(ax^2+bx+c\), where a, b, and c are constants. In this case, the trinomial \(x^2-2x-15\) is already in the standard form. Note the values of a, b, and c: a=1, b=-2, c=-15.
02

Finding factors

The goal here is to find two numbers that multiply to give ac (which is just c in this case, because a is 1) and add to give b. We're looking for two numbers that multiply to -15 and add to -2. The numbers -5 and 3 fit this criteria because -5 * 3 = -15 and -5 + 3 = -2.
03

Write the factor form

Replace the trinomial with the pair of binomials found in step 2. This means that \(x^2-2x-15\) factors into \((x - 5)(x + 3)\).
04

Check the solution

Generally, it is good practice to check the solution. You can do this by re-expanding the pair of binomials to see if you get back to the given trinomial. \((x - 5)(x + 3)\) expands to \(x^{2}-2x-15\), confirming that the factored form is accurate.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ 5^{-2}>2^{-5} $$

Simplify each expression. Assume that all variables represent positive numbers. $$ \left(49 x^{-2} y^{4}\right)^{-\frac{1}{2}}\left(x y^{\frac{1}{2}}\right) $$

Place the correct symbol, \(>\) or \(<,\) in the shaded area between the given numbers. Do not use a calculator. Then check your result with a calculator. a. \(3^{\frac{1}{2}} \quad 3^{\frac{1}{3}}\) b. \(\sqrt{7}+\sqrt{18} \sqrt{7+18}\)

How do the whole numbers differ from the natural numbers?

Factor and simplify each algebraic expression. $$-8(4 x+3)^{-2}+10(5 x+1)(4 x+3)^{-1}$$
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