Problem 14 Divide and check. $$\left(16 x... [FREE SOLUTION]

Chapter 5: Problem 14

Divide and check. $$\left(16 x^{6}+32 x^{5}-8 x^{2}\right) \div\left(-8 x^{2}\right)$$

Short Answer

Expert verified

-2x^{4} - 4x^{3} + 1

Step by step solution

Set Up the Division

Write the expression as a fraction showing the division of the polynomial by the monomial: \[ \frac{16x^{6} + 32x^{5} - 8x^{2}}{-8x^{2}} \]

Divide Each Term

Divide each term in the numerator by the term in the denominator individually: \[ \frac{16x^{6}}{-8x^{2}}, \frac{32x^{5}}{-8x^{2}}, \text{and} \frac{-8x^{2}}{-8x^{2}} \]

Simplify Each Division

Simplify each of the terms:\[ \frac{16x^{6}}{-8x^{2}} = -2x^{4}, \frac{32x^{5}}{-8x^{2}} = -4x^{3}, \text{and} \frac{-8x^{2}}{-8x^{2}} = 1 \]

Combine Simplified Terms

Put all the simplified terms together to form the final answer: \[ -2x^{4} - 4x^{3} + 1 \]

Check the Solution

To check, multiply the simplified answer by the divisor and see if it matches the original polynomial: \[-8x^{2}(-2x^{4} - 4x^{3} + 1) = 16x^{6} + 32x^{5} - 8x^{2}\]. This matches the original polynomial, so the solution is correct.

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Key Concepts

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

monomial division

Monomial division involves dividing a polynomial by a monomial. It's like breaking down a complex task into smaller, manageable steps. In our exercise, we need to divide $(16x^{6} + 32x^{5} - 8x^{2}) \div (-8x^{2}) $.
To accomplish this, we'll split the division into separate parts. Each term in the polynomial numerator is divided by the monomial denominator. These terms are:

\frac{16x^{6}}{-8x^{2}}
\frac{32x^{5}}{-8x^{2}}
\frac{-8x^{2}}{-8x^{2}}

Breaking down this division makes the problem more manageable and clearer.

simplifying expressions

Simplifying expressions is key to making math problems easier to solve. By breaking down complex expressions, we can strip away the extra 'layers' and see the core components. In our example, after dividing each term, we have:

\texpression{ \frac{16x^{6}}{-8x^{2}} = -2x^{4}
\frac{32x^{5}}{-8x^{2}} = -4x^{3}
\frac{-8x^{2}}{-8x^{2}} = 1 }

This process transforms the original, more complicated polynomial into $-2x^{4} - 4x^{3} + 1$. This result is simpler and easier to understand.

checking solutions

Checking solutions is an important step to ensure your work is correct. It gives you confidence that you've solved the problem accurately. To verify, you rework the problem in reverse. For the division problem, multiply your final simplified result $-2x^{4} - 4x^{3} + 1$ by the original divisor $-8x^{2}$.

This means:
$-8x^{2} \times (-2x^{4} - 4x^{3} + 1) = 16x^{6} + 32x^{5} - 8x^{2} $. This matches the original polynomial. So, our solution of $-2x^{4} - 4x^{3} + 1$ is correct. Always remember to check your solutions to confirm their accuracy.

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91影视

Divide and check. $$\left(16 x^{6}+32 x^{5}-8 x^{2}\right) \div\left(-8 x^{2}\right)$$

Short Answer

Step by step solution

Set Up the Division

Divide Each Term

Simplify Each Division

Combine Simplified Terms

Check the Solution

Key Concepts

monomial division

simplifying expressions

checking solutions

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