Problem 213 Multiply. Use either method. $... [FREE SOLUTION]

91影视

Prealgebra

Lynn Marecek, MaryAnne Anthony-Smith

$Math Studyset 91影视 Explanations$ Math

2016 Edition

Chapter 10: Problem 213

Multiply. Use either method. $$(2 x+1)\left(x^{2}-5 x-6\right)$$

Short Answer

Expert verified

The product is $2x^3 - 9x^2 - 17x - 6$.

Step by step solution

- Distribute Terms

Distribute each term in the first polynomial $2x + 1$ to every term in the second polynomial $x^{2} - 5x - 6$

- Multiply 2x by Each Term

First, multiply $2x $ by each term in $x^{2} - 5x - 6$. That is $2x \times x^2$, $2x \times -5x$, and $2x \times -6$.$2x \times x^2 = 2x^3$$2x \times -5x = -10x^2$$2x \times -6 = -12x$

- Multiply 1 by Each Term

Next, multiply $1 $ by each term in $x^{2} - 5x - 6$. That is $1 \times x^2$, $1 \times -5x$, and $1 脳 -6$.$1 \times x^2 = x^2$$1 \times -5x = -5x $$1 脳 -6 = -6 $

- Combine Like Terms

Now add all the resulting terms from Steps 2 and 3 together:$2x^3 + (-10x^2) + (-12x) + x^2 + (-5x) + (-6) $. Combine like terms:$2x^3 + (-10x^2 + x^2) + (-12x - 5x) + (-6) $The simplified expression is \2x^3 - 9x^2 - 17x - 6 \.

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Key Concepts

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

Distribute Terms

Distributing terms is the process of multiplying each term in one polynomial by every term in another polynomial. Imagine you鈥檙e handing out items to each person in a line. Every person gets one item from each set of items you have. In our case, we need to distribute the terms in $2x + 1$ to every term in $x^{2} - 5x - 6$. This means:

Multiplying $2x$ by each term in $x^{2} - 5x - 6$
And multiplying $1$ by each term in $x^{2} - 5x - 6$

Let鈥檚 break that down step-by-step so it鈥檚 crystal clear:

Combining Like Terms

After distributing the terms, we end up with a bunch of different terms. Combining like terms means adding or subtracting terms that have the same variable raised to the same power. For example, $-10x^2$ and $x^2$ are like terms because they both have $x^2$.
Here鈥檚 how to combine like terms for our exercise:

First, collect all the $x^3$ terms (though there is only one in this case, $2x^3$).
Next, gather all the $x^2$ terms: $-10x^2$ and $x^2$, which adds up to $-9x^2$.
Then, collect all the $x$ terms: $-12x$ and $-5x$, giving us $-17x$.
Finally, the constant term is simply $-6$.

Adding them all together, we get the combined polynomial.

Simplifying Polynomial Expressions

Simplifying polynomial expressions involves reducing them down to their simplest form by combining like terms and ensuring there are no unnecessary parentheses.
Here is the final step:

From the previous steps, we have the terms: $2x^3, -10x^2, x^2, -12x, -5x, -6$.
Combine the like terms to simplify: $2x^3 - 9x^2 - 17x - 6$.

Always double-check your work to make sure you didn鈥檛 miss any terms or make any arithmetic mistakes.
After combining and simplifying, our final polynomial is as simple as it can get!

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91影视

Multiply. Use either method. $$(2 x+1)\left(x^{2}-5 x-6\right)$$

Short Answer

Step by step solution

- Distribute Terms

- Multiply 2x by Each Term

- Multiply 1 by Each Term

- Combine Like Terms

Key Concepts

Distribute Terms

Combining Like Terms

Simplifying Polynomial Expressions

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