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Chapter 0: Problem 36

Multiply and simplify. $$ \left(2 z^{3}+5 u^{2}\right)\left(7 z^{3}-u^{2}\right) $$

Short Answer

Expert verified
14z^6 + 33z^3u^2 - 5u^4

Step by step solution

01

- Distribute the First Term

Distribute the first term of the first polynomial, which is 2z^3, to each term of the second polynomial. Calculate:\( 2z^3 \times 7z^3 = 14z^6 \)\( 2z^3 \times (-u^2) = -2z^3u^2 \)
02

- Distribute the Second Term

Distribute the second term of the first polynomial, which is 5u^2, to each term of the second polynomial. Calculate:\( 5u^2 \times 7z^3 = 35z^3u^2 \)\( 5u^2 \times (-u^2) = -5u^4 \)
03

- Combine Like Terms

Combine the results of distribution to get the polynomial in a simplified form:\( 14z^6 + 35z^3u^2 - 2z^3u^2 - 5u^4 \)Then, combine the like terms:\( 14z^6 + (35z^3u^2 - 2z^3u^2) - 5u^4 = 14z^6 + 33z^3u^2 - 5u^4 \)

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Key Concepts

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

distributive property
When multiplying polynomials, one essential concept is the Distributive Property. This property allows us to distribute, or multiply, each term in one polynomial by each term in the other polynomial.

Let's break it down with an example. Consider the multiplication problem \( (2z^3 + 5u^2)(7z^3 - u^2) \). To solve this, you need to distribute each term in the first polynomial to each term in the second polynomial:

    \(2z^3 \times 7z^3 = 14z^6\)
    \(2z^3 \times (-u^2) = -2z^3u^2\)

      After distributing 2z^3, we move on to the next term in the first polynomial, which is 5u^2:

        \(5u^2 \times 7z^3 = 35z^3u^2\)
        \(5u^2 \times (-u^2) = -5u^4\)

        These steps use the Distributive Property by ensuring every term in the first polynomial interacts with every term in the second polynomial.
combining like terms
After distributing, we move to the next vital step: Combining Like Terms. Like terms are terms that contain the same variables raised to the same power.

From our distribution result: \( 14z^6 + 35z^3u^2 - 2z^3u^2 - 5u^4 \), we have terms that can be combined.

Very simply, this means:
    Any terms with \(z^6\) will join together.
    Any terms with \(z^3u^2\) will combine.
    Any terms with \(u^4\) will stay as they are if there are no like terms.

    In our example, we should add/subtract the coefficients of like terms while keeping the variables and their exponents intact.

      Combine \(35z^3u^2 - 2z^3u^2\): it yields \(33z^3u^2\).

      So, the expression becomes: \(14z^6 + 33z^3u^2 - 5u^4\).
simplifying expressions
Finally, we reach Simplifying Expressions. This step ensures the polynomial is in its simplest form, meaning combining all possible like terms and presenting the expression in a streamlined manner.

From the previous section, we had the expression: \(14z^6 + 33z^3u^2 - 5u^4\).

This intermediate result already combines like terms, so no further work is needed there. However, let's summarize:

    Check each term for common factors.
    Ensure all like terms are combined.
    Verify the expression can't be simplified further.

    This ensures the polynomial is presented in its simplest, most understandable form.

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Most popular questions from this chapter

Write the expanded form for \((a+b)^{2}\).

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