Chapter 0: Problem 95
Find the quotient and the remainder. Check your work by verifying that Quotient \(\cdot\) Divisor \(+\) Remainder \(=\) Dividend $$ 5 x^{4}-3 x^{2}+x+1 \text { divided by } x^{2}+2 $$
Short Answer
Expert verified
Quotient: \( 5x^2 - 13 \), Remainder: \( x + 27 \).
Step by step solution
01
- Set up the long division
Write down the dividend \( 5x^4 - 3x^2 + x + 1 \) inside the division symbol and the divisor \( x^2 + 2 \) outside.
02
- Divide the leading terms
Divide the leading term of the dividend \( 5x^4 \) by the leading term of the divisor \( x^2 \). This gives \( 5x^2 \). Write \( 5x^2 \) above the division symbol.
03
- Multiply and subtract
Multiply \( 5x^2 \) by \( x^2 + 2 \) which results in \[ 5x^4 + 10x^2 \]. Write this product under the corresponding terms of the dividend and subtract: \[ (5x^4 - 3x^2) - (5x^4 + 10x^2) = -13x^2 \].
04
- Bring down the next term
Bring down the next term of the dividend, which is \(x \), to get \[ -13x^2 + x \].
05
- Repeat the division
Divide the new leading term \( -13x^2 \) by \( x^2 \). This gives \( -13 \). Write \( -13 \) above the division symbol.
06
- Multiply and subtract again
Multiply \( -13 \) by \( x^2 + 2 \), resulting in \[ -13x^2 - 26 \]. Subtract this product from \[ -13x^2 + x + 1 \]: \[ (-13x^2 + x + 1) - (-13x^2 - 26) = x + 27 \].
07
- Write the quotient and remainder
The quotient is \[ 5x^2 - 13 \] and the remainder is \[ x + 27 \].
08
- Verify the result
Verify by checking if \[ (5x^2 - 13)(x^2 + 2) + (x + 27) = 5x^4 - 3x^2 + x + 1 \]. After expansion and simplification, you will find that the equality holds true.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Dividend and Divisor
In polynomial long division, the terms 'dividend' and 'divisor' are essential to understand.
The **dividend** is the polynomial you want to divide. In this case, our dividend is \(5x^4 - 3x^2 + x + 1\).
The **divisor** is the polynomial you are dividing by. Here, it is \(x^2 + 2\).
To set up the problem, we place the dividend inside the long division symbol (also known as the division bracket) and the divisor outside it. This setup allows us to systematically divide each term of the dividend by the leading term of the divisor, step by step.
The key to understanding this concept is to recognize that dividing polynomials is very similar to dividing numbers, but involves more algebraic manipulation.
The **dividend** is the polynomial you want to divide. In this case, our dividend is \(5x^4 - 3x^2 + x + 1\).
The **divisor** is the polynomial you are dividing by. Here, it is \(x^2 + 2\).
To set up the problem, we place the dividend inside the long division symbol (also known as the division bracket) and the divisor outside it. This setup allows us to systematically divide each term of the dividend by the leading term of the divisor, step by step.
The key to understanding this concept is to recognize that dividing polynomials is very similar to dividing numbers, but involves more algebraic manipulation.
Quotient and Remainder
When performing polynomial long division, the goal is to decompose the dividend into a series of divisions that yields a quotient and a remainder.
The **quotient** is the result of the division, and it represents how many times the divisor can be completely factored out of the dividend. In our problem, the quotient is \(5x^2 - 13\).
The **remainder** is what's left over after the division process is complete. It represents the part of the dividend that cannot be evenly divided by the divisor. For our problem, the remainder is \(x + 27\).
Understanding this step-by-step breakdown is crucial:
The **quotient** is the result of the division, and it represents how many times the divisor can be completely factored out of the dividend. In our problem, the quotient is \(5x^2 - 13\).
The **remainder** is what's left over after the division process is complete. It represents the part of the dividend that cannot be evenly divided by the divisor. For our problem, the remainder is \(x + 27\).
Understanding this step-by-step breakdown is crucial:
- First, you divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient.
- Next, you multiply the whole divisor by this term and subtract the product from the original dividend to form a new polynomial.
- Then, you repeat the process with the new polynomial until you can't divide anymore.
Verification of Results
To ensure that our division is correct, we need to verify the results. In mathematical terms, this involves checking that the equation \((\text{Quotient} \times \text{Divisor}) + \text{Remainder} = \text{Dividend}\) holds true.
For our problem, we calculated the quotient as \(5x^2 - 13\) and the remainder as \(x + 27\). We need to show that \((5x^2 - 13)(x^2 + 2) + (x + 27) = 5x^4 - 3x^2 + x + 1\).
Here's how you can verify:
Everything checks out, which means our quotient and remainder are correct! This verification step is essential to confirm the accuracy of your polynomial division.
For our problem, we calculated the quotient as \(5x^2 - 13\) and the remainder as \(x + 27\). We need to show that \((5x^2 - 13)(x^2 + 2) + (x + 27) = 5x^4 - 3x^2 + x + 1\).
Here's how you can verify:
- First, expand the product \((5x^2 - 13)(x^2 + 2)\).
- After expanding, you get \(5x^4 + 10x^2 - 13x^2 -26 = 5x^4 - 3x^2 -26\).
- Next, add the remainder, which is \(x + 27\).
- Combine the terms: \(5x^4 - 3x^2 -26 + x + 27\).
- Simplify the expression to get back to the original dividend \(5x^4 - 3x^2 + x + 1\).
Everything checks out, which means our quotient and remainder are correct! This verification step is essential to confirm the accuracy of your polynomial division.