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Chapter 0: Problem 29

Factor each trinomial, or state that the trinomial is prime. $$4 x^{2}+16 x+15$$

Short Answer

Expert verified
The factored form of the trinomial \(4x^2 + 16x + 15\) is \((2x + 3)(2x + 5)\).

Step by step solution

01

Identify the coefficients

Identify the coefficients and the constant in the trinomial. In the given trinomial \(4x^2 + 16x + 15\), \(a = 4\), \(b = 16\), and \(c = 15\).
02

Find product of \(a\) and \(c\)

Calculate the product of \(a\) and \(c\), which gives \(4 \cdot 15 = 60\). Now, look for two numbers that multiply to 60 and add up to 16.
03

Identify the factors

The two numbers that fulfill these requirements are 6 and 10 because \(6 \cdot 10 = 60\) and \(6 + 10 = 16\). These two numbers will be used to split the middle term.
04

Rewrite the middle term and factor by grouping

Rewrite the trinomial \(4x^2 + 16x + 15\) as \(4x^2 + 6x + 10x + 15\). Now, group the terms to factor by grouping, so we have \(2x(2x + 3) + 5(2x + 3)\).
05

Final factorization

Since both terms have a common factor of \(2x + 3\), factor it out, leaving the factored form of the trinomial as \((2x + 3)(2x + 5)\).

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Most popular questions from this chapter

List all numbers from the given set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, f. real numbers. $$\\{-5,-0 . \overline{3}, 0, \sqrt{2}, \sqrt{4}\\}$$

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