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Chapter 0: Problem 26

Factor each trinomial, or state that the trinomial is prime. $$3 x^{2}-2 x-5$$

Short Answer

Expert verified
The factorized form of the trinomial \(3x^{2} - 2x - 5\) is \((3x - 5)(x + 1)\)

Step by step solution

01

Identify the Coefficients and Constant

The trinomial is \(3x^{2} - 2x - 5\). Here, the coefficient of \(x^{2}\) is 3, the coefficient of x is -2, and the constant term is -5.
02

Find the Product of the Coefficients and Constant

The product of the coefficient of \(x^{2}\) (3) and the constant term (-5) is -15.
03

Find Two Numbers that Multiply to -15 and Add to -2

The two numbers that meet these conditions are -5 and 3, because -5 times 3 equals -15, and -5 plus 3 equals -2.
04

Rewrite the Trinomial Using these numbers

We can break down the term -2x using the numbers -5 and 3. This leads to: \(3x^{2} - 5x + 3x - 5\)
05

Group and Factor

Now group the terms and factor by grouping: \(x(3x - 5) + 1(3x - 5)\) Now, take out the common binomial factor, which is \(3x - 5\), So the factored form of the trinomial is: \((3x - 5)(x + 1)\)

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Most popular questions from this chapter

a. A mathematics professor recently purchased a birthday cake for her son with the inscription $$\text { Happy }\left(2^{\frac{5}{2}} \cdot 2^{\frac{3}{4}} \div 2^{\frac{1}{4}}\right) \text { th Birthday. }$$ How old is the son? b. The birthday boy, excited by the inscription on the cake, tried to wolf down the whole thing. Professor Mom, concerned about the possible metamorphosis of her son into a blimp, exclaimed, "Hold on! It is your birthday, so why not take \(\frac{8^{-\frac{4}{3}}+2^{-2}}{16^{-\frac{3}{4}}}\) of the cake? I'll eat half of what's left over." How much of the cake did the professor eat?

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$-x-x=-x+(-x)=0$$

Explain how to factor \(3 x^{2}+10 x+8\)

Factor Completely. $$x^{4}-10 x^{2} y^{2}+9 y^{4}$$

Factor completely. $$(x-5)^{-\frac{1}{2}}(x+5)^{-\frac{1}{2}}-(x+5)^{\frac{1}{2}}(x-5)^{-\frac{3}{2}}$$
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