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Chapter 3: Problem 7

Find the quotient and remainder if \(f(x)\) is divided by \(p(x)\). $$f(x)=10 x+4 ; \quad p(x)=2 x-5$$

Short Answer

Expert verified
Quotient is 5 and remainder is 29.

Step by step solution

01

Set up the division

First, we'll write the division in a polynomial long division format:\[ \frac{10x + 4}{2x - 5} \] This sets up our format for polynomial division, where \(f(x) = 10x + 4\) and \(p(x) = 2x - 5\).
02

Find the first term of the quotient

Divide the leading term of \(f(x)\), which is \(10x\), by the leading term of \(p(x)\), which is \(2x\). \[ \frac{10x}{2x} = 5 \]So, the first term of our quotient is 5.
03

Multiply and subtract

Multiply \(5\) by the entire divisor \((2x - 5)\):\[ 5 \times (2x - 5) = 10x - 25 \]Subtract the result from \(10x + 4\):\[ (10x + 4) - (10x - 25) = 29 \] So we subtract to find the remainder, and our polynomial becomes the constant 29.
04

Write the quotient and remainder

The quotient obtained is 5 and the remainder is 29. Since the degree of the remainder (a constant) is less than the degree of the divisor \((2x - 5)\), we have completed the division with:\[ f(x) = (2x - 5)(5) + 29 \]

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Key Concepts

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

Quotient and Remainder
When dividing polynomials, the outcome is expressed in terms of a quotient and a remainder. This is similar to the division you might do with whole numbers but adapted to work with polynomials.
  • The quotient is the result of dividing the polynomials, excluding the leftover part (remainder).
  • The remainder is what is left over after the division process is completed - it has a lower degree than the divisor.
For instance, when dividing polynomial \(f(x) = 10x + 4\) by \(p(x) = 2x - 5\), we obtain a quotient of \(5\) and a remainder of \(29\). The result can be written as: \[f(x) = (2x - 5)(5) + 29\]
This expression captures the division perfectly, where: - \(5\) is the quotient,- \(29\) is the remainder.
Polynomial Long Division
Polynomial long division is a systematic method used to divide one polynomial by another. It follows steps remarkably similar to long division of numbers. Here's how it works:
  • Setup: Arrange the polynomials to identify the dividend and the divisor. In our example, \(f(x) = 10x + 4\) is the dividend and \(p(x) = 2x - 5\) is the divisor.

  • Divide: Compare the leading term of the dividend with the leading term of the divisor. Here we divide \(10x\) by \(2x\) to get \(5\), the first term of our quotient.

  • Multiply: Multiply the entire divisor by the first term obtained. So, \(5\times(2x - 5)\) results in \(10x - 25\).

  • Subtract: Subtract the product from the original dividend to simplify. \[(10x + 4) - (10x - 25) = 29\]

  • The result shows what remains, which is the remainder.
We've now divided our polynomials using the long division technique, resulting in a quotient of \(5\) and a remainder of \(29\).
Degree of a Polynomial
Understanding the degree of a polynomial is crucial when performing polynomial division. The degree indicates the highest power of the variable present in the polynomial. Why should we care?
  • The degree is key to defining when the division process is complete. For division, when your remainder鈥檚 degree is less than that of the divisor, division is finished.
  • In our example: The divisor \(2x - 5\) is a polynomial of degree \(1\) (because the highest power of \(x\) is \(x^1\)), and our remainder \(29\) is a constant (having degree \(0\)).

This match between the remainder degree and the expected condition ensures that our division process was done correctly. Understanding and verifying these degrees allow us to confidently assert that our solution: \[f(x) = (2x - 5)(5) + 29\] is accurate.

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