Question

Find the sum of \(a, b, c,\) and \(d\) if \(\frac{x^{3}-2 x^{2}+3 x+5}{x+2}=a x^{2}+b x+c+\frac{d}{x+2}\)

Short answer

The sum is -9.

Step-by-step solution

- Set Up Polynomial Long Division

To find the sum of the coefficients and the remainder, set up the polynomial long division: divide the numerator \(x^3 - 2x^2 + 3x + 5\) by the denominator \(x + 2\).

- Divide the First Term

Divide the first term of the numerator \(x^3\) by the first term of the denominator \(x\) to get \(x^2\). Write this below the division line.

- Multiply and Subtract

Multiply \(x^2\) by \(x + 2\) which gives \(x^3 + 2x^2\). Subtract this from the original numerator: \((x^3 - 2x^2 + 3x + 5) - (x^3 + 2x^2)\) resulting in \(-4x^2 + 3x + 5\).

- Repeat Division Process

Divide \(-4x^2\) by \(x\) to get \(-4x\). Multiply \(-4x\) by \(x + 2\) which gives \(-4x^2 - 8x\). Subtract this from \(-4x^2 + 3x + 5\): \((-4x^2 + 3x + 5) - (-4x^2 - 8x)\) resulting in \(11x + 5\).

- Final Division and Subtraction

Divide \(11x\) by \(x\) to get \(11\). Multiply \(11\) by \(x + 2\) which gives \(11x + 22\). Subtract this from \(11x + 5\): \((11x + 5) - (11x + 22)\) resulting in \(-17\).

- Identify Coefficients and Remainder

From the division, the quotient is \(ax^2 + bx + c = x^2 - 4x + 11\) and the remainder is \(d = -17\). Thus, \(a = 1\), \(b = -4\), \(c = 11\), and \(d = -17\).

- Sum the Coefficients

Calculate the sum of \(a, b, c,\) and \(d\): \(1 + (-4) + 11 + (-17) = -9\).

Key concepts

Polynomial Division

Polynomial division is similar to the long division you learned in your earlier math classes, but instead of numbers, we are working with polynomials. We take a polynomial (called the dividend) and divide it by another polynomial (called the divisor). The objective is to express the dividend as the product of the divisor and a quotient polynomial plus a remainder. Here's an easy-to-follow method for polynomial division:

• Set up the division: Write the dividend and divisor.
• Divide the leading term of the dividend by the leading term of the divisor to get the first term of the quotient.
• Multiply this term by the entire divisor and subtract the result from the original dividend.
• Repeat the steps with the resulting polynomial until the degree of the remainder is less than the degree of the divisor.

This approach is fundamental in many algebraic processes, including simplifying expressions and solving polynomial equations. Specifically, to solve the exercise, the division helps us express the original polynomial in a simpler form plus a remainder.

Remainder Theorem

The Remainder Theorem is a useful shortcut when evaluating polynomials. It states: Given a polynomial, if you divide it by \(x - c\), the remainder of that division is the same as evaluating the polynomial at \(c\). In simpler terms, if we have a polynomial \(P(x)\) and we want to find \(P(c)\), instead of performing the entire polynomial division, we can directly substitute \(c\) into the polynomial. For example, consider we have the polynomial \(P(x) = x^3 - 2x^2 + 3x + 5\) and we divide it by \(x + 2\). According to the theorem, the remainder of this division would be \(P(-2)\). This helps in confirming the results of polynomial division methods and can speed up problem-solving significantly.

Coefficient Sum

The coefficient sum involves adding up all the numerical coefficients of a polynomial. In our example, once we have identified the coefficients \(a, b, c\), and \(d\) through polynomial division, we can sum them simply to find the answer. Here is how we do it:

• Identify the coefficients from the quotient of the division. In our problem, they are identified as \(a = 1, b = -4, c = 11\).
• Include the remainder as the \(d\) value (here, \(d = -17\)).
• Sum these identified values: 1 + (-4) + 11 + (-17).

Thus, we find the coefficient sum to be \(-9\), which is the final answer.

Algebraic Expressions

Understanding algebraic expressions is key to solving polynomial problems. An algebraic expression includes variables, constants, and algebraic operations like addition, subtraction, multiplication, and division. In polynomials, these expressions often represent equations in which we need to find solutions or simplifications. For example, in our polynomial \(x^3 - 2x^2 + 3x + 5\), each term (\textaround{x^3, -2x^2}) plays a role. Here's a structured way to break these down:

• Identify different terms and their respective coefficients.
• Understand the power (or degree) each variable holds, indicating how many times it multiplies itself.
• Learn operations between terms and how each contributes to solving the equation through processes like polynomial division or evaluating via the Remainder Theorem.

Mastering algebraic expressions is crucial because they form the backbone of more complex algebra and calculus topics, setting a strong foundation for all higher-level math studies.