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Chapter 5: Problem 26

Find each product. $$ \begin{array}{r} {-r^{2}-4 r+8} \\ {3 r-2} \\ \hline \end{array} $$

Short Answer

Expert verified
-3r^3 - 10r^2 + 32r - 16

Step by step solution

01

Distribute each term

Multiply each term in the first polynomial \(-r^2 - 4r + 8\) by the first term in the second polynomial \(3r\). Then, multiply each term in the first polynomial by the second term in the second polynomial \(-2\).
02

Multiply-term by term

Perform the multiplications: \((3r \times -r^2) = -3r^3\), \((3r \times -4r) = -12r^2\), and \(3r \times 8 = 24r\). Then, \((-2 \times -r^2) = 2r^2\), \((-2 \times -4r) = 8r\), and \((-2 \times 8) = -16\).
03

Combine like terms

Combine all the results from Step 2: \(-3r^3\), \(-12r^2 + 2r^2\), \((24r + 8r)\), and \-16. The combined polynomial is \(-3r^3 - 10r^2 + 32r - 16\).

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Key Concepts

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

distributive property
The distributive property is a fundamental concept in algebra that allows us to simplify expressions involving polynomials.
When multiplying polynomials, you use the distributive property to multiply each term in one polynomial by each term in another polynomial.
This means that every term from the first polynomial must be multiplied by every term in the second polynomial.
For instance, to multiply \(-r^2 - 4r + 8\) by \(3r - 2\), you would:
  • Multiply \(-r^2\) by each term in \((3r - 2)\)
  • Then, multiply \(-4r\) by each term in \((3r - 2)\)
  • Finally, multiply \(8\) by each term in \((3r - 2)\)
You end up with all these intermediate terms that need to be combined: \((3r \times -r^2)= -3r^3, (3r \times -4r)= -12r^2, (3r \times 8)= 24r\) and \((-2 \times -r^2)= 2r^2, (-2 \times -4r) = 8r, (-2 \times 8)= -16\).
combining like terms
Combining like terms is an essential step in simplifying polynomial expressions.
It involves adding or subtracting terms that have the same variable raised to the same power.
In our example, after distributing, we get several terms: \(-3r^3, -12r^2, 24r, 2r^2, 8r, -16\).
We need to combine the terms with the same powers:
  • The cubic term is \(-3r^3\), which has no like terms to combine with.
  • The quadratic terms are \(-12r^2\) and \(2r^2\), which add up to \(-10r^2\).
  • The linear terms are \(24r\) and \(8r\), which add up to \(32r\).
  • The constant term is \(-16\), which also stands alone.
By combining these like terms, you simplify the expression to \(-3r^3 - 10r^2 + 32r - 16\).
multiplying polynomials
Multiplying polynomials is a process that involves using the distributive property multiple times and then combining like terms.
Here鈥檚 a step-by-step guide:
  • Write both polynomials in standard form if they are not already.
  • Use the distributive property to ensure that every term in the first polynomial multiplies every term in the second polynomial.
  • Multiply each pair of terms.
  • Write down each resulting term with its coefficient and variable correctly indicated.
  • Combine the like terms to simplify your expression.
Let's summarize it using our example.
Multiply each term in \(-r^2 - 4r + 8\) by each term in \(3r - 2\), like this: \((3r \times -r^2) = -3r^3\), \(3r \times -4r = -12r^2\), \(3r \times 8 = 24r\), \((-2 \times -r^2) = 2r^2\), \((-2 \times -4r) = 8r\), and \(-2 \times 8 = -16\).
Bringing them together: \[-3r^3 - 12r^2 + 24r + 2r^2 + 8r - 16 \], and then combining like terms: \[-3r^3 - 10r^2 + 32r - 16\].
This results in the simplified expression: \(-3r^3 - 10r^2 + 32r - 16\).

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