Problem 70 Multiply. $$8(x+6)(2 x+1)$$... [FREE SOLUTION]

Chapter 6: Problem 70

Multiply. $$8(x+6)(2 x+1)$$

Short Answer

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The short answer to the question 'Multiply $8(x+6)(2 x+1)$' is $16x^2 + 104x + 48$.

Step by step solution

Multiply 8 and (x+6)

To do this, we need to apply the distributive property by multiplying 8 with each term in the expression $(x+6)$: $8(x+6) = 8 \times x + 8 \times 6 = 8x + 48$ Now we have the expression: $(8x + 48)(2x + 1)$

Multiply (8x + 48) and (2x + 1)

Now, we are going to apply the distributive property again using the FOIL method (First, Outer, Inner, Last) to multiply the two terms in $(8x + 48)$ with the two terms in $(2x + 1)$: First: $8x \times 2x = 16x^2$ Outer: $8x \times 1 = 8x$ Inner: $48 \times 2x = 96x$ Last: $48 \times 1 = 48$ Now we need to add the results from these multiplications: $16x^2 + 8x + 96x + 48$

Combining like terms

We have two like terms to combine: $8x$ and $96x$. Adding those terms, we get: $16x^2 + 104x + 48$ The final simplified expression is: $16x^2 + 104x + 48$

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Key Concepts

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

Polynomial Multiplication

Polynomial multiplication involves multiplying two or more expressions that contain variables raised to different powers. While it might seem complex, it simply extends the rules of arithmetic multiplication to polynomial expressions. Here's how you can think about it:

Each term in one polynomial is multiplied by every term in the other polynomial.
The degree of the final product is usually the sum of the degrees of the factors being multiplied.
After multiplying, you'll often end up with several terms, which may include terms that can be combined.

In the exercise example, we first multiplied 8 by $x+6$, using the distributive property, which gave us $8x + 48$. Then, we needed to multiply the resulting expression $8x + 48$ by $2x + 1$. Thus, in polynomial multiplication, each term must interact with every other term from the polynomials involved.

FOIL Method

The FOIL method is a straightforward way to multiply two binomials and stands for First, Outer, Inner, and Last terms. This method systematically ensures that all terms are accounted for. Let's break it down further:

First: Multiply the first terms in each binomial.
Outer: Multiply the outer terms in the product.
Inner: Multiply the inner terms in the product.
Last: Multiply the last terms in each binomial.

In our step-by-step solution, we used the FOIL method to handle $8x + 48$ and $2x + 1$. It gave us the results $16x^2$ from the first terms, $8x$ from the outer terms, $96x$ from the inner terms, and $48$ from the last terms. Each multiplication is crucial for obtaining the complete polynomial before combining terms.

Combining Like Terms

Once you've expanded your polynomial through multiplication, the next task is simplifying it by combining like terms. Like terms are those that have the same variables raised to the same power. Here鈥檚 how you can combine them:

Identify terms that can be added together. These are the ones with identical variable components.
Add or subtract the coefficients of these like terms.
Simplify the expression to its most compact form.

In our example, after using the FOIL method, we were left with $16x^2 + 8x + 96x + 48$. By combining the like terms $8x$ and $96x$, the expression simplifies to $16x^2 + 104x + 48$. Combining like terms helps in making the expression cleaner and easier to understand.

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Multiply. $$8(x+6)(2 x+1)$$

Short Answer

Step by step solution

Multiply 8 and (x+6)

Multiply (8x + 48) and (2x + 1)

Combining like terms

Key Concepts

Polynomial Multiplication

FOIL Method

Combining Like Terms

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