Problem 93 - College Algebra Essentials

Question

Will help you prepare for the material covered in the next section. $$\text { Multiply: }(7-3 x)(-2-5 x)$$

Step-by-Step Solution

Answer

The product of the multiplication of the two binomial expressions \((7-3x)(-2-5x)\) is \(15x^2-29x-14\)

Step 1: Expand the first term

Begin the process by multiplying the first term of the first binomial with both the terms of the second binomial. So, \(7(-2)\) and \(7(-5x)\) gives \(-14\) and \(-35x\) respectively. Write it in a way that the term with a higher degree comes first. So, \(-35x-14\).

Step 2: Expand the second term

Next, multiply the second term of the first binomial with both the terms of the second binomial. So, \(-3x(-2))\) and \(-3x(-5x))\) gives \(6x\) and \(15x^2\) respectively. Here, \(15x^2\) is the higher degree term. So, it comes before \(6x)\), making it, \(15x^2+6x)\.

Step 3: Sum up all terms

The final step is to add the expanded terms from Step 1 and Step 2. So, sum up \(-35x-14\) and \(15x^2+6x)\) results in \(15x^2-29x-14)\.

Key Concepts

Binomial Expansion

When you encounter problems involving binomials like \((7-3x)(-2-5x)\), you're stepping into the realm of binomial expansion! Binomial expansion means taking two terms (binomials) and expanding them into a longer expression by multiplying them together. Each term of one binomial is multiplied by each term of the second binomial.

In our exercise, we start with \((7-3x)(-2-5x)\). Here, we have two binomials, and our goal is to use multiplication to expand them into a four-term polynomial. The expanded result should represent the sum of all those multiplied terms. It’s somewhat like unfolding or unwrapping the binomials to see what lies beneath!

In binomial expansion, always keep track of each multiplication and make sure every term from the first binomial gets paired with every term in the second binomial.

Distributive Property

The distributive property is like having a party where one number wants to make sure everyone gets a share. If you have an expression like \(a(b+c)\), the distributive property helps you spread \(a\) over both \(b\) and \(c\). This means rewriting it as \(a \cdot b + a \cdot c\).

In the context of our exercise, we use the distributive property twice! First, with the 7 from the binomial \((7-3x)\) and second with the \(-3x\). Each term from \((7-3x)\) takes a turn multiplying every term from \((-2-5x)\).

• First, distribute the 7 to the \(-2\) and \(-5x\), giving us \(-14\) and \(-35x\).
• Then, distribute the \(-3x\) to \(-2\) and \(-5x\), giving \(6x\) and \(15x^2\).

The distributive property ensures every term gets appropriately expanded, providing a foundation to later combine like terms.

Combining Like Terms

Once you’ve expanded your binomials using the distributive property, your next task is to combine like terms. This reduces the expression to its simplest form. Like terms are terms that have the same variable raised to the same power.

Look at the expanded expression: \(15x^2 + 6x - 35x - 14\). Here, the \(x^2\) term stands alone because there are no other \(x^2\) terms to combine it with.

• For the \(x\) terms \((6x\) and \(-35x)\), combine them by adding their coefficients: \(6 - 35 = -29\).
• The constant \(-14\) remains unchanged as it is already on its own.

Combining these, you get the final polynomial: \(15x^2 - 29x - 14\). By combining like terms, the expression becomes succinct and easily interpretable. This crucial step ensures your final polynomial accurately represents the initial multiplication of the binomials.