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Chapter 10: Problem 549

Multiply the binomials using various methods. $$(y-4)(y+12)$$

Short Answer

Expert verified
The product is \( y^2 + 8y - 48 \).

Step by step solution

01

- Apply the Distributive Property

Distribute each term in the first binomial to every term in the second binomial. You need to compute each of the following: 1. Distribute the term \( y \) in the first binomial to the second binomial \( (y+12) \). 2. Distribute the term \( -4 \) in the first binomial to the second binomial \( (y+12) \).
02

- Multiply Each Term

Compute the following multiplications: 1. \( y \cdot y = y^2 \)2. \( y \cdot 12 = 12y \)3. \( -4 \cdot y = -4y \)4. \( -4 \cdot 12 = -48 \)
03

- Combine Like Terms

Add all the results from Step 2 together and combine the like terms: \[ y^2 + 12y - 4y - 48 \]Combine \( 12y \) and \(-4y \): \[ y^2 + 8y - 48 \]

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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 key when multiplying binomials like \( (y-4)(y+12) \). It allows us to break down the process into smaller, more manageable parts.
To apply the Distributive Property, we distribute each term in the first binomial to every term in the second binomial. This means:
  • First, take \( y \) from \( (y-4) \) and multiply it by each term in \( (y+12) \).
  • Then, take \( -4 \) from \( (y-4) \) and multiply it by each term in \( (y+12) \).
By following these steps, we can systematically compute each part of the expression, which forms the foundation of our solution.
Combining Like Terms
Once we have distributed each term and performed the multiplications, we end up with several terms that need to be combined.
In our example, we had the terms: \ y^2 + 12y - 4y - 48 \. To simplify this, we combine the like terms.
  • Like terms are terms that have the same variable raised to the same power. Here, \( 12y \) and \( -4y \) are like terms because both are coefficients of \( y \).
  • Add or subtract the coefficients of these like terms. So, \( 12y - 4y = 8y \).
This process helps us simplify the expression to its most reduced form, yielding \ y^2 + 8y - 48 \.
Polynomial Multiplication
Polynomial multiplication involves multiplying two or more polynomials together. In our case, we are multiplying two binomials. By following the Distributive Property, we ensure that every possible combination of terms between the two binomials is multiplied.
Here's how it works step-by-step:
  • Multiply the first term of the first binomial by each term in the second binomial.
  • Multiply the second term of the first binomial by each term in the second binomial.
To illustrate with our example \( (y-4)(y+12) \):
  • Multiply \( y \) with \( y \) yielding \ y^2 \.
  • Multiply \( y \) with \( 12 \) yielding \ 12y \.
  • Multiply \( -4 \) with \( y \) yielding \ -4y \.
  • Finally, multiply \( -4 \) with \( 12 \) yielding \ -48 \.
Once all terms are computed, we combine them in the final step to get the simplified polynomial: \ y^2 + 8y - 48 \.

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Most popular questions from this chapter

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