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Chapter 5: Problem 37

Find each product. $$-b\left(a^{2}-a b+b^{2}\right)$$

Short Answer

Expert verified
The product \(-b(a^{2}-ab+b^{2})\) when simplified becomes \(-b^{3} - ba^{2} + ab^{2}\).

Step by step solution

01

Distributive property

The first step is to use the distributive property to distribute the -b to all the terms inside the brackets. Hence, \(-b(a^{2}-ab+b^{2}) = -ba^{2} + b^{2}a - b^{3}\).
02

Rearranging terms

After distributing -b to the terms in the bracket, we end up with \(-ba^{2} + b^{2}a - b^{3}\). For a standardized expression, rearrange the terms to group together the different powers of variables, in decreasing power order. The final result would then be \(-b^{3} - ba^{2} + ab^{2}\).

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Key Concepts

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

Polynomial Multiplication
Polynomial multiplication involves multiplying each term in one polynomial by each term in another polynomial. This exercise features the Distributive Property, which involves distributing one element across the terms inside the parentheses. The general form can be written as:
  • Each term outside the parentheses is multiplied with every term inside.
  • Sum the results to get the final expression.
For example, when we have a term like \[-b\left(a^{2}-ab+b^{2}\right),\]each term inside the bracket must be multiplied by \(-b\). Thus, we compute \(-b \cdot a^{2}, -b \cdot (-ab),\) and \(-b \cdot b^{2}\) in sequence.
Negative Coefficient
When dealing with negative coefficients, attention is required, as a negative sign affects the sign of the resulting product. In this case, our coefficient is \(-b\). Handling negative coefficients involves:
  • Preserving the negative signs during multiplication.
  • Reversing signs according to the rules of multiplying negatives.
For example:
  • Multiply \(-b\) times \(a^{2}\), leading to \(-ba^{2}\).
  • Multiply \(-b\) times \(-ab\), which yields a positive result \(b^{2}a\).
  • Finally, \(-b \cdot b^{2} = -b^{3}\).
Notice how each multiplication is executed carefully maintaining the correct signs.
Rearranging Terms
Rearranging terms is a strategy used to present a polynomial in a neat and standardized form. This is specially important in algebra to recognize and compare equivalent expressions easily. In this exercise:
- The expression after distribution was \(-ba^{2} + b^{2}a - b^{3}\).- We rearrange these terms to reflect a conventional order, which often means placing terms in decreasing order of their degree.
For instance, in our solution:
  • Start with \(-b^{3}\) as it is the term with the highest exponent on a single variable.
  • Next comes \(-ba^{2}\).
  • Finally, \(+ab^{2}\) is written at the last.
This conventional order helps in further mathematical manipulations or comparisons.
Algebraic Expression Simplification
Simplification in algebra is the process of condensing expressions into simpler forms by combining like terms and ensuring terms are in their simplest form. An important part of simplification includes:
  • Using the Distributive Property effectively.
  • Making sure all like terms are combined where possible, although this example does not have like terms.
  • Arranging them in a standard order, which helps in seeing any potential further simplifications.
In simple terms, every term after distribution must stand alone unless it can be mixed with another similar term, achieving the simplest possible expression from given terms. This refinement ensures clarity and readability, which is vital in understanding algebraic expressions comprehensively.

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Most popular questions from this chapter

What is the degree of a polynomial? Provide an example with your explanation.

perform the indicated operations. $$\begin{aligned} &\left[\left(4 y^{2}-3 y+8\right)-\left(5 y^{2}+7 y-4\right)\right]-\left[\left(8 y^{2}+5 y-7\right)+\left(-10 y^{2}+4 y+3\right)\right] \end{aligned}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. When a certain polynomial is divided by \(2 x+4,\) the quotient is $$x-3+\frac{17}{2 x+4}$$ What is the polynomial?

After dividing a polynomial by a binomial, explain how to check the answer.

Exercises \(172-174\) will help you prepare for the material covered in the first section of the next chapter. In each exercise, find the product. $$(x+3)\left(x^{2}+5\right)$$
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