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Chapter 6: Problem 30

Factor completely. If a polynomial is prime, state this. $$ 2 t^{2}-19-6 t $$

Short Answer

Expert verified
The expression factors to \[ -x^{3}(x - 1)(x - 3) \].

Step by step solution

01

Identify the Greatest Common Factor (GCF)

Look at each term: - -x^{5}, - 4x^{4}, - 3x^{3}. Find the greatest common factor (GCF) of each term. The GCF here is - x^{3}.
02

Factor out the GCF

Factor -x^{3} from each term: \[ -x^{5} + 4x^{4} - 3x^{3} = -x^{3}(x^{2} - 4x + 3) \].
03

Factor the quadratic expression

The expression x^{2} - 4x + 3 can be factored further. Look for two numbers that multiply to 3 and add up to -4. These numbers are -1 and -3: \[ x^{2} - 4x + 3 = (x - 1)(x - 3) \].
04

Combine the factors

Combine the factors from steps 2 and 3: \[ -x^{3}(x - 1)(x - 3) \].

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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 factoring polynomials, the first step is often to identify the Greatest Common Factor (GCF). The GCF is the largest factor that divides all terms in the polynomial. For example, in the polynomial -x^{5} + 4x^{4} - 3x^{3}, each term has a factor of -x^{3}.
This means that the GCF is -x^{3}. By factoring out the -x^{3} from each term, we simplify the polynomial. This helps in further steps of polynomials factorization.
Remember:
  • Look at the coefficients (numbers part).
  • Look at the variables and their exponents.
In our example: for coefficients (-1, 4, -3), the GCF is 1. For variables (x^{5}, x^{4}, x^{3}), the GCF is x^{3}, combined: -x^{3}.
factoring quadratics
Factoring quadratics is the process of breaking down a quadratic expression into a product of two binomials. After finding the GCF, the next step often involves factoring a quadratic polynomial. Here, we have a quadratic polynomial (x^{2} - 4x + 3).
The goal is to find two numbers that multiply to the last term (3) and add up to the middle term鈥檚 coefficient (-4). In this case, those numbers are -1 and -3, because:
  • The product of -1 and -3 is 3.
  • The sum of -1 and -3 is -4.
Thus, the quadratic polynomial can be written as: (x - 1)(x - 3). This process of factoring quadratic expressions is used frequently in algebra to simplify and solve equations.
polynomial factorization
Polynomial factorization involves breaking down a polynomial into simpler components that, when multiplied together, give the original polynomial. This process is critical for simplifying algebraic expressions and solving polynomial equations.
In our example, after factoring out the GCF (- x^{3}) and then factoring the quadratic (x^{2} - 4x + 3), we combine all factors to write the polynomial in its completely factored form:
-x^{3}(x - 1)(x - 3).
Steps for polynomial factorization are:
  • Identify the GCF.
  • Factor out the GCF.
  • Factor any remaining quadratic expressions.
  • Combine all factors to get the completely factored polynomial.
Polynomial factorization makes complex polynomials more manageable and is a fundamental technique in algebra.

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Most popular questions from this chapter

Determine whether each of the following is a difference of squares. $$ -1+64 t^{2} $$

Solve. Round any irrational solutions to the nearest thousandth. $$ x^{3}+3 x^{2}+x-1=0 $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ x^{3}-7 x^{2}-4 x+28 $$

Factor completely. Remember to look first for a common factor. If a polynomial is prime, state this. $$ x^{2}+2 x y+y^{2}-9 $$

Review multiplying binomials using FOIL $$ (x-2)(x+7) $$
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