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Chapter 6: Problem 53

$$\text { In Exercises } 43-66, \text { factor completely.}$$ $$4 x^{3}+12 x^{2}-72 x$$

Short Answer

Expert verified
The completely factored form of the given polynomial \(4x^3 + 12x^2 - 72x\) is \(4x(x-3)(x+6)\).

Step by step solution

01

Identify Common Factors

Notice that all terms of the polynomial \(4x^3 + 12x^2 - 72x\) have a common factor of \(4x\). Factoring out \(4x\) gives \(4x(x^2 + 3x -18)\).
02

Factor the Quadratic Expression

The quadratic expression \(x^2 + 3x - 18\) can be factorized into \((x-3)(x+6)\), observing that \(-3\) and \(6\) are the roots of the equation \(x^2 +3x -18 = 0\). Substituting \(x-3\) and \(x+6\) for \(x^2 + 3x - 18\) in the factored expression from step 1 results in \(4x(x-3)(x+6)\).

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Key Concepts

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

Common Factors
Factoring polynomials often starts with identifying common factors. These are factors that are shared by each term in the polynomial. In the expression \(4x^3 + 12x^2 - 72x\), the common factor is \(4x\). It is found by examining coefficients for numerical commonalities and the smallest power of \(x\) present across all terms.
Removing the common factor simplifies the expression significantly. Extracting \(4x\) from each term yields \(4x(x^2 + 3x - 18)\).
  • Check the coefficients in each term for a greatest common divisor.
  • Look for the lowest power of any variables shared by the terms.
Recognizing and factoring out common factors is an essential first step in polynomial factorization. It reduces the complexity and makes further factorization easier.
Quadratic Expression
A quadratic expression follows the general form \(ax^2 + bx + c\). Understanding how to work with these expressions is a key aspect of polynomial factorization. In this example, the targeted quadratic expression is \(x^2 + 3x - 18\).
Quadratics can have complex factorizations depending on their coefficients. Trinomial expressions like this are simpler as they typically break down into two binomials. In this case, it can be factorized into \((x-3)(x+6)\).
  • Finding two numbers that multiply to \(c\) and add to \(b\) is crucial.
  • Using these, rewrite the quadratic expression as a binomial product.
Mastery of these skills allows for quick identification and manipulation of quadratic expressions in algebra.
Roots
Roots of a polynomial are values of \(x\) where the polynomial equals zero. Finding these is vital in factorizing expressions since they offer insights into the structure of the polynomial. For the quadratic equation \(x^2 + 3x - 18 = 0\), solving gives us \(-3\) and \(6\) as roots.
These roots indicate the values for which each binomial component of the quadratic equals zero. For this expression:
  • Roots \(-3\) and \(6\) signal respective binomial factors \((x + 3)\) and \((x - 6)\).
  • The equation is factored as \((x - 3)(x + 6)\).
Understanding roots can simplify the factorization process greatly and are foundational to solving and simplifying polynomial equations.
Polynomial Factorization
Polynomial factorization remains a central technique in simplifying algebraic expressions and solving equations. It involves breaking down a complex polynomial into simpler, multipliable factors.
Our original polynomial, \(4x^3 + 12x^2 - 72x\), was simplified through:
  • Identifying and factoring out the common factor \(4x\).
  • Further breaking down the resulting quadratic expression \(x^2 + 3x - 18\) into \((x - 3)(x + 6)\).
After these steps, the complete factorization of the polynomial is \(4x(x-3)(x+6)\). This process not only simplifies the polynomial but is instrumental in solving equations and understanding deeper algebraic concepts. Factorization helps reveal properties that are less obvious in an expanded expression.

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Most popular questions from this chapter

Use factoring to solve quadratic equation. Check by substitution or by using a graphing utility and identifying \(x\)-intercepts. \(4 y^{2}+44 y+121=0\)

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