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Chapter 6: Problem 82

Factor by grouping. \(10 t^{3}-2 t^{2} s^{2}-5 t s+s^{3}\)

Short Answer

Expert verified
The factored form is \( (2t^2 - s)(5t - s^2) \).

Step by step solution

01

Group the terms

Divide the polynomial into groups of two terms each: \( (10t^3 - 2t^2 s^2) + (-5ts + s^3) \)
02

Factor out the greatest common factor (GCF) from each group

From the first group \(10t^3 - 2t^2 s^2\), the GCF is \(2t^2\), so factor it out: \( 2t^2(5t - s^2) \)From the second group \(-5ts + s^3\), the GCF is \(s\), so factor it out: \( s(-5t + s^2) \)
03

Simplify and rearrange

Notice that the terms inside the parentheses are similar, but not identical due to the sign difference. We can factor out \(-1\) from the second term to match them: \( 2t^2(5t - s^2) - s(-5t + s^2) = 2t^2(5t - s^2) - s(s^2 - 5t) \)
04

Factor by grouping

Combine the expressions by factoring out the common binomial \((5t - s^2)\): \( (2t^2 - s)(5t - s^2) \)

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Key Concepts

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

greatest common factor (GCF)
The greatest common factor, often abbreviated as GCF, is the largest factor that divides two or more numbers. When we talk about factoring polynomials, the GCF is the highest degree of the variable that can be factored out from all terms in a polynomial. Consider the polynomial from our exercise, which is:
10t^3 - 2t^2s^2 - 5ts + s^3.
To factor by grouping, we first look at each pair of terms separately. For the first group, 10t^3 - 2t^2s^2, we find that both terms contain t^2 and a common number factor of 2. Hence, the GCF here is 2t^2.
For the second group, -5ts + s^3, each term contains at least one s. Thus, the GCF for this group is s. Finding the GCF in each group makes the polynomial easier to factor further.
polynomial
A polynomial is an expression consisting of variables, coefficients, and exponents, which are combined using addition, subtraction, and multiplication. In the given problem, the polynomial is: 10t^3 - 2t^2s^2 - 5ts + s^3. This polynomial has four terms and involves two variables, t and s.
To factor polynomials by grouping, follow these steps:
  • Divide the polynomial into groups.

  • Factor out the GCF from each group.

  • Simplify and look for common binomials.

  • Factor out the common binomial from each term.

In our example, once we grouped the terms and factored out the GCF from each group, we noticed the common binomial terms, which allowed us to factor the polynomial by grouping.
binomial
A binomial is a polynomial with exactly two terms. For example, in the expression 5t - s^2, we have two terms: 5t and - s^2. The same binomial appears in both groups after factoring out the GCF: 2t^2(5t - s^2) and s(-5t + s^2).
Upon observing that -5t + s^2 is essentially the same as -(5t - s^2), we can factor by grouping.

Steps to factor by grouping using binomials:

  • Refactor the binomials to match one another if needed, by factoring out a common factor, including negative ones.

  • Once you have matching binomials, factor them out.

  • Combine the terms to rewrite the polynomial in its factored form.

In our problem, factoring by grouping gives the final expression: (2t^2 - s)(5t - s^2).

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