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Chapter 10: Problem 28

Multiply the polynomials. $$(w-3)(z-9)$$

Short Answer

Expert verified
The result is \( wz - 9w - 3z + 27 \).

Step by step solution

01

Use the distributive property

To multiply \( (w-3)(z-9) \), use the distributive property to expand the expression: \( w(z-9) - 3(z-9) \).
02

Distribute the terms

Distribute each term in the first polynomial by each term in the second polynomial: \( wz - 9w - 3z + 27 \).
03

Simplify the expression

Combine the terms to get the final simplified polynomial: \( wz - 9w - 3z + 27 \).

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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 multiply a single term by each term within a parenthesis.
It is commonly expressed as:
  • For any three numbers a, b, and c:
For instance, if you have

  • In our given exercise, we need to apply the distributive property to multiply (w-3) by (z-9).
      • Remember when applying distributive property in polynomials, ensure every term in the first polynomial is distributed across every term in the second polynomial. This step-by-step approach makes sure no term is left out.
Polynomial Expansion
Expanding polynomials entails using the distributive property to rewrite the polynomial in a simpler and more manageable form. It allows us to break down complex polynomials into simpler parts.
Here's how to expand the polynomials:
    On applying the distributive property, multiply each term in one polynomial by each term in the other.
    In our exercise, we distribute (w-3) by (z-9) as:
    • w(z-9) - 3(z-9).
    Distributing the Terms
    • Start with the first term in the first polynomial, w. Multiply w by each term in the second polynomial.
    • This gives us wz - 9w.
    • Now, take the second term in the first polynomial, -3. Multiply it by each term in the second polynomial.
    • This results in -3z + 27.
    After polynomial expansion, we write down all the terms: wz - 9w - 3z + 27.
Simplifying Expressions
Once we've expanded our polynomial, the next step involves simplifying the expression. Simplifying helps in combining like terms and reducing the polynomial to its most basic form.
To simplify polynomials,
  • Look for like terms (terms with the same variables raised to the same power).
  • In our example, the expanded form is wz - 9w - 3z + 27.
      • Combining Like Terms
      • Since there are no like terms in wz, -9w, -3z, and +27, we don't need to combine any terms.
      • In this case, our simplified polynomial remains as wz - 9w - 3z + 27.

      This process ensures that our polynomial is as simplified and manageable as possible, making it easier to understand and work with in future calculations.

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    Most popular questions from this chapter

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