Problem 2 - 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 \(P(x)=D(x) \cdot Q(x)+R(x)\). \(P(x)=x^{3}+4 x^{2}-6 x+1, \quad D(x)=x-1\)

Step-by-Step Solution

Answer

The division is perfect with quotient \( x^2 + 5x - 1 \) and remainder 0.

Step 1: Setting Up Synthetic Division

Identify the divisor and set up the synthetic division. For synthetic division with \( x - c \), use \( c = 1 \) from \( D(x) = x - 1 \). The coefficients for \( P(x) \) are \( [1, 4, -6, 1] \). Place 1 outside the synthetic division setup.

Step 2: Perform Synthetic Division

Write the leading coefficient of \( P(x) \), which is 1, in the lower row. Multiply this by 1 (the divisor) and write the result under the next coefficient. Add the two numbers, and continue this process sequentially for all coefficients. - Start with 1: Place it in the bottom row. - Multiply 1 by 1: Write 1 under the second coefficient (4). - Add: \( 4+1=5 \), place 5 in the bottom row. - Multiply 5 by 1: Write 5 under the next coefficient (-6). - Add: \( -6+5=-1 \), place -1 in the bottom row. - Multiply -1 by 1: Write -1 under the last coefficient (1). - Add: \( 1-1=0 \), place 0 in the bottom row.

Step 3: Interpret the Result

The numbers on the bottom row represent the coefficients of the quotient polynomial \( Q(x) \) and the remainder \( R(x) \). From left to right, they are: 1, 5, -1, and 0. Thus, \( Q(x) = x^2 + 5x - 1 \) and \( R(x) = 0 \).

Step 4: Express the Desired Equation

Using the results from synthetic division, express the original polynomial \( P(x) \) in the desired form: \( P(x) = D(x) \cdot Q(x) + R(x) \). Substituting the values, we have: \( x^3 + 4x^2 - 6x + 1 = (x - 1)(x^2 + 5x - 1) + 0 \).

Key Concepts

Synthetic Division

Synthetic division is a simplified method to divide a polynomial by a binomial of the form \(x - c\). It's much faster and more efficient than the traditional long division, especially when dealing with polynomials. You begin by identifying \(c\) from your divisor \(D(x) = x - c\). In our example, \(D(x) = x - 1\), so \(c = 1\). The key steps are:

• Write down the coefficients of the polynomial \(P(x)\), which for this example are \([1, 4, -6, 1]\).
• Place \(c\) outside the synthetic division setup.
• Bring the first coefficient directly down as it is.
• Multiply this number by \(c\) and add it to the next coefficient. Repeat this process for all coefficients.

This division method allows us to directly find the polynomial quotient \(Q(x)\) and the remainder \(R(x)\) without fully writing out all the terms, making your calculations quicker and neater.

Remainder Theorem

The Remainder Theorem helps to easily find the remainder of a polynomial division without completing the division. If a polynomial \(P(x)\) is divided by \(x - c\), then the remainder of this division is \(P(c)\). In synthetic division, the remainder is the final number in the bottom row after completing the procedure.

In the provided solution, after performing synthetic division, the remainder \(R(x)\) turns out to be 0. Thus, according to the Remainder Theorem, \(P(1) = 0\). This indicates that \(D(x) = x - 1\) is a factor of \(P(x)\). In polynomial problems, understanding the Remainder Theorem can save you time and effort and help in determining factorization quickly.

Polynomial Quotient

The quotient polynomial \(Q(x)\) is what you get after performing the division, excluding the remainder. When dividing polynomials, finding the quotient helps you express the original polynomial in a simplified form: \(P(x) = D(x) \cdot Q(x) + R(x)\).

In our example, synthetic division gave us the bottom-row coefficients, which represent \(Q(x)\). Here, it's \(x^2 + 5x - 1\). Importantly, having a remainder of 0, as in our exercise, means the division is exact, and \(P(x)\) can be perfectly expressed as the product of \(D(x)\) and \(Q(x)\) without any remainder.

Long Division in Algebra

Long division in algebra is analogous to long division with numbers, extended to polynomial expressions. It's a more traditional way of dividing polynomials compared to synthetic division. If the divisor is not of the form \(x - c\), then long division is typically used.

• Start by dividing the leading term of the dividend by the leading term of the divisor.
• Multiply the entire divisor by this result and subtract it from the original dividend.
• Repeat this process with the new polynomial that results after subtraction.

Though more labor-intensive than synthetic division, it’s versatile and can handle a wider variety of divisor forms. Notably, with practice, long division can become just as resolvable and is essential for understanding division beyond simple cases.