Problem 3 - Algebra and Trigonometry

Question

Two polynomials \(P\) and \(D\) are given. Use either synthetic or long division to divide \(P(x)\) by \(D(x),\) and express \(P\) in the form \(P(x)=D(x) \cdot Q(x)+R(x)\). \(P(x)=2 x^{3}-3 x^{2}-2 x, \quad D(x)=2 x-3\)

Step-by-Step Solution

Answer

The division yields: \(P(x) = (2x - 3)(2x^2 - 2) + 3\).

Step 1: Set Up Components for Division

Identify the dividend and the divisor. Here, the dividend polynomial is \(P(x) = 2x^3 - 3x^2 - 2x\) and the divisor polynomial is \(D(x) = 2x - 3\). We are going to use synthetic division for simplicity since we have a linear divisor.

Step 2: Adjust Division Method Choice

Since we are using synthetic division and \(D(x) = 2x - 3\) is linear, we first find the zero of \(D(x)\), which is \(x = \frac{3}{2}\). We'll use this value in the process.

Step 3: Arrange and Begin Synthetic Division

Write down the coefficients of \(P(x)\), which are \([2, -3, -2, 0]\). Notice the zero for the missing constant term. On the left, write the zero of \(D(x)\), which is \(\frac{3}{2}\).

Step 4: Perform Synthetic Division

1. Bring down the leading coefficient 2. 2. Multiply \(\frac{3}{2}\) by 2 and write 3 below the next coefficient.3. Add \(-3 + 3 = 0\).4. Multiply \(\frac{3}{2}\) by 0 and add to the next coefficient \(-2\), resulting in \(-2\).5. Multiply \(\frac{3}{2}\) by \(-2\) and add to 0, resulting in 3.

Step 5: Conclude with Quotient and Remainder

The quotient coefficients are \([2, 0, -2]\), which gives \(Q(x) = 2x^2 - 2\). The remainder is \(R(x) = 3\). Therefore, the expression of \(P(x)\) in the form of \(P(x) = D(x) \cdot Q(x) + R(x)\) is: \(P(x) = (2x - 3)(2x^2 - 2) + 3\).

Step 6: Verify the Result

Expand \((2x - 3)(2x^2 - 2)\) and add the remainder: Calculate \(4x^3 - 6x^2 - 4x + 6 + 3\) which simplifies to \(2x^3 - 3x^2 - 2x\), confirming that the division is correct.

Key Concepts

Synthetic Division

Synthetic division is a simplified method for dividing polynomials, specifically when the divisor is of the form \(x - c\). This technique requires less writing and is faster, making it a popular choice for students. To perform synthetic division, follow these steps:

• Find the zero of the divisor. For \(D(x) = 2x - 3\), solve for \(x\) to get \(x = \frac{3}{2}\).
• Arrange the coefficients of the polynomial \(P(x)\) in descending order. Here, \(P(x) = 2x^3 - 3x^2 - 2x\) has coefficients \([2, -3, -2, 0]\).
• Perform the synthetic division process by bringing down the leading coefficient, multiplying, and adding sequentially.

Make sure to place a zero for any missing term, as it simplifies the process and ensures accuracy. Synthetic division quickly helps find the quotient \(Q(x)\) and remainder \(R(x)\) of the polynomial division.

Long Division

Long division, akin to the division method used in arithmetic, can divide any polynomials regardless of the degree or type of the divisor. It might be preferred when working with more complex divisors instead of the synthetic substitution.While it involves more steps than synthetic division, its universal applicability makes it a must-know skill.

• Set up by writing the dividend \(P(x)\) beneath the long division symbol and the divisor \(D(x)\) outside it.
• Divide the leading term of the dividend by the leading term of the divisor to find the first term of the quotient.
• Multiply the entire divisor by this new term and subtract the result from the dividend to get a new polynomial.
• Repeat the process until reaching a remainder whose degree is less than the divisor's.

This method allows you to evaluate the polynomial, extracting both the quotient and remainder while offering a deeper understanding of polynomial behaviors during division.

Remainder Theorem

The Remainder Theorem provides a quick way to find the remainder of a polynomial \(P(x)\) when divided by a linear divisor \(x-c\). This theorem states that the remainder of the division of \(P(x)\) by \(x-c\) is simply \(P(c)\). Here's how you apply it:

• Identify \(c\) from the divisor \(D(x) = 2x - 3\), thus \(c = \frac{3}{2}\).
• Substitute \(c\) into the polynomial \(P(x)\): Calculate \(P(\frac{3}{2})\).

Once computed, the result is your remainder.This method not only checks the remainder we found using synthetic or long division but also provides an efficient way to quickly verify computations without redoing the entire division.

Quotient and Remainder

When dividing any polynomial \(P(x)\) by another \(D(x)\), the result can be expressed in the standard form \(P(x) = D(x) \cdot Q(x) + R(x)\), where \(Q(x)\) is the quotient and \(R(x)\) is the remainder. Here are a few key points about the quotient and remainder:

• The degree of \(Q(x)\) is always less than the degree of \(P(x)\).
• The degree of \(R(x)\) is less than the degree of \(D(x)\), ensuring the validity of the division.
• If the division has no remainder, this means \(P(x)\) is exactly divisible by \(D(x)\).

The results are useful in factoring polynomials and solving polynomial equations, as they simplify and redefine polynomial expressions in a manageable manner.