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Chapter 6: Problem 37

Factor the trinomial completely. $$ 3 x^{3}+18 x^{2}+24 x $$

Short Answer

Expert verified
The completely factored form of the trinomial \(3x^3+18x^2+24x\) is \(3x(x+2)(x+4)\)

Step by step solution

01

Find the Greatest Common Factor (GCF)

Looking at the given trinomial \(3x^3+18x^2+24x\), the first step in factoring is to find the Greatest Common Factor (GCF) of these terms. In this case, it's simply \(3x\). This means that every term in the trinomial can be divided evenly by \(3x\). Doing so gives us: \(3x(x^2 + 6x + 8)\)
02

Factor the trinomial inside the brackets

The second step involves further factoring the remaining trinomial within the brackets. To factor \(x^2+6x+8\) completely, one needs to look for two numbers that multiply to 8 (the last term) and whose sum is 6 (the coefficient of the middle term). The numbers 2 and 4 meet this criteria, so the trinomial can be factored as: \(3x(x+2)(x+4)\)
03

Verify the factorized form

The final step is to verify the factored form. One does this by expanding the trinomial to see if it matches the original trinomial. By expanding, we get: \(3x*(x+2)*(x+4)\ =3x^3+18x^2+24x\), which is the same as the original trinomial. Therefore, the factored form is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Factor (GCF)
When working with algebraic expressions, identifying the Greatest Common Factor (GCF) is a crucial first step in the factoring process. The GCF is the largest factor that divides all terms in the expression evenly. In the context of the trinomial \(3x^3 + 18x^2 + 24x\), the GCF is \(3x\). This means:
  • All coefficients can be divided by 3.
  • All terms have at least one \(x\) present.
Finding the GCF simplifies the trinomial into smaller parts. By factoring out the GCF \(3x\), you're left with \(3x(x^2 + 6x + 8)\). This simplification makes it easier to further manipulate or solve the expression. Spotting the GCF helps reduce complex equations to more manageable sections.
Polynomial Factoring
Polynomial factoring involves breaking down a polynomial into simpler 'factors' that, when multiplied together, yield the original polynomial. After removing the GCF from \(3x(x^2 + 6x + 8)\), the next focus is factoring the remaining trinomial \(x^2 + 6x + 8\).
  • Look for two numbers that multiply to the constant term (8) and add up to the middle coefficient (6).
  • In this case, 2 and 4 meet the criteria because \(2 \times 4 = 8\) and \(2 + 4 = 6\).
This trinomial can thus be expressed as \((x + 2)(x + 4)\), forming the factorization of the original expression into \(3x(x+2)(x+4)\). Polynomial factoring is an important skill for solving equations and simplifying expressions, often used to reveal roots or zeroes of polynomials.
Algebraic Expressions
Algebraic expressions like the trinomial \(3x^3 + 18x^2 + 24x\) are combinations of variables and constants combined using arithmetic operations. These expressions can often be simplified or rewritten to highlight certain properties or make them easier to work with.
  • Factoring reduces complex terms into simpler components.
  • It helps in solving equations, simplifying forms, and finding zeros.
Understanding how to manipulate and factor algebraic expressions is fundamental in algebra. It allows you to uncover relationships between variables, solve for specific values, or simply reveal the inherent structure of the mathematical model you're working with. Whether tackling quadratic equations or simplifying complex terms, factoring is an invaluable tool in the algebraic toolkit.

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Most popular questions from this chapter

A diver jumps from a diving board that is 32 feet above the water with an initial velocity of 16 feet per second (see figure). The height \(h\) (in feet) of the diver is modeled by the position equation $$ h=-16 t^{2}+16 t+32 $$ where \(t\) is the time measured in seconds. How long will it take for the diver to reach the water?

Use the Zero-Factor Property to solve the equation. $$ (y-2)(y-3)=0 $$

In Exercises \(1-6\), use the Zero-Factor Property to solve the equation. $$ x(x-5)=0 $$

Solve the equation. $$ (x-5)(x-1)=21 $$

Solve the equation and check your solution.xplain how to identify and factor a perfect square trinomial. $$ \frac{3}{4}(12 x-8)=10 $$
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