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Chapter 5: Problem 73

Find product. \((2 y-7)(y+4)\)

Short Answer

Expert verified
The product is \(2y^2 + y - 28\).

Step by step solution

01

Distribute the First Term

Multiply the first term in the first binomial by each term in the second binomial. This means you will multiply \(2y \) by \(y+4\).This will give you \(2y \times y + 2y \times 4 = 2y^2 + 8y\).
02

Distribute the Second Term

Next, multiply the second term in the first binomial by each term in the second binomial. This means you will multiply \(-7\) by \(y+4\).This will give you \(-7 \times y + (-7) \times 4 = -7y - 28\).
03

Combine Like Terms

Now, add the results of the two distributions together: \(2y^2 + 8y - 7y - 28\). Combine the like terms, which are \(8y\) and \(-7y\).This results in \(2y^2 + (8y - 7y) - 28 = 2y^2 + y - 28\).

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Key Concepts

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

Binomial Expansion
Binomial expansion involves multiplying two binomials together to find the product. A binomial is an algebraic expression with two terms, like \(a + b\). When we expand binomials, we're essentially applying multiplication rules to each term in the first binomial with each term in the second binomial. For the exercise given, the binomials are \( (2y - 7) \) and \( (y + 4) \). The expansion process means you'll end up with terms formed from each possible combination of terms in the two binomials.
To keep track of all these multiplications, the distributive property comes in handy. Let's move forward to discuss this property and how it's applied.
Distributive Property
The distributive property directs us to multiply every term inside a binomial by each term in the other binomial. It's represented as \(a(b + c) = ab + ac\). This property ensures each term in one binomial is effectively distributed through the other binomial.

In our exercise:
\[ (2y-7)(y+4) \]
We first distribute \( 2y \) (the first term in the first binomial) to each term in \( (y + 4) \).
This gives us \[ 2y \times y + 2y \times 4 = 2y^2 + 8y \]
Next, distribute \(-7\) (the second term in the first binomial) to each term in \( (y + 4) \).
This produces \(-7 \times y + -7 \times 4 = -7y - 28 \)
Combining all these terms will lead us into our next concept: combining like terms.
Combining Like Terms
Combining like terms is the process of adding or subtracting terms that have the same variable and the same exponent. This step simplifies the polynomial to its final form.

In the exercise:
After distributing, we are left with the terms \( 2y^2 \), \( 8y \), \(-7y\), and \(-28\).
To combine these like terms, focus on the coefficients and keep the variable part unchanged:
  • \(+8y\) and \(-7y\) are like terms, so we combine them:
    \8y - 7y = 1y, or simply y
  • The other terms, \( 2y^2 \) and \(-28\), do not have matching variable parts with others, so they remain as is.
Thus, the simplified result of the multiplication is:
\[ 2y^2 + y - 28 \]
Understanding these key steps鈥攂inomial expansion, using the distributive property, and combining like terms鈥攎akes polynomial multiplication a much smoother process.

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Most popular questions from this chapter

Factor each trinomial completely. $$ 18 x^{2}-48 x y+32 y^{2} $$

Apply the special factoring rules of this section to factor each binomial or trinomial. $$ y^{2}-1.4 y+0.49 $$

Students often become confused as to how to handle a constant, such as 2 in the equation \(2 x(3 x-4)=0 .\) How would you explain to someone how to solve this equation and how to handle the constant \(2 ?\)

Factor completely. \(2 x^{6}+8 x^{5}-42 x^{4}\)

Apply the special factoring rules of this section to factor each binomial or trinomial. $$ m^{2}+\frac{2}{3} m+\frac{1}{9} $$
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