Question

Perform each division. Express the result in the form \(\frac{P(x)}{x-a}=Q(x)+\frac{R}{x-a} .\) Identify any restrictions on the variable. a) \(\left(x^{3}+7 x^{2}-3 x+4\right) \div(x+2)\) b) \(\frac{11 t-4 t^{4}-7}{t-3}\) c) \(\left(x^{3}+3 x^{2}-2 x+5\right) \div(x+1)\) d) \(\left(4 n^{2}+7 n-5\right) \div(n+3)\) e) \(\frac{4 n^{3}-15 n+2}{n-3}\) f) \(\left(x^{3}+6 x^{2}-4 x+1\right) \div(x+2)\)

Short answer

(a) \( \frac{x^3 + 7x^2 - 3x + 4}{x + 2} = x^2 + 5x - 13 + \frac{30}{x + 2}\) and \(x eq -2\).

Step-by-step solution

Understand the Problem

Given each polynomial division, express the result in the form \(\frac{P(x)}{x-a} = Q(x) + \frac{R}{x-a}\). Identify any restrictions on the variable. Use polynomial long division to achieve this.

Example (a) - Polynomial Division Setup

Divide \(x^3 + 7x^2 - 3x + 4 \) by \(x + 2\): Set up the division with the divisor \(x + 2\) outside the division symbol and the dividend \(x^3 + 7x^2 - 3x + 4\) inside.

Perform the Division (a)

1. Divide the leading term \(x^3\) by \(x\) to get the first term of the quotient: \(x^2\).2. Multiply \(x + 2\) by \(x^2\): \(x^3 + 2x^2\).3. Subtract \(x^3 + 2x^2\) from \(x^3 + 7x^2 - 3x + 4\) to get the new polynomial: \(5x^2 - 3x + 4\).4. Repeat this process.5. Divide \(5x^2\) by \(x\) to get \(5x\).6. Multiply \(x + 2\) by \(5x\): \(5x^2 + 10x\).7. Subtract \(5x^2 + 10x\) from \(5x^2 - 3x + 4\) to get \( -13x + 4\).8. Divide \(-13x\) by \(x\) to get \(-13\).9. Multiply \(x + 2\) by \(-13\): \(-13x - 26\).10. Subtract \(-13x - 26\) from \(-13x + 4\) to get \(30\).11. The quotient is \(x^2 + 5x - 13\) and the remainder is \(30\). So, \(\frac{x^3 + 7x^2 - 3x + 4}{x + 2} = x^2 + 5x - 13 + \frac{30}{x + 2}\).

Identify Restrictions (a)

Identify any restrictions: The denominator \(x + 2\) cannot be zero, so \(x eq -2\).

Example (b) - Polynomial Division Setup

Divide \(11t - 4t^4 - 7 \) by \(t - 3\): Set up the division with the divisor \(t - 3\) outside the division symbol and the dividend \(- 4t^4 + 0t^3 + 0t^2 + 11t - 7\) inside.

Perform the Division (b)

1. Divide the leading term \(-4t^4\) by \(t\) to get the first term of the quotient: \(-4t^3\).2. Multiply \(t - 3\) by \(-4t^3\): \(-4t^4 + 12t^3\).3. Subtract \(-4t^4 + 12t^3\) from \(- 4t^4 + 0t^3 + 0t^2 + 11t - 7\) to get the new polynomial: \(-12t^3 + 0t^2 + 11t - 7\).4. Repeat the process. The next term is obtained by dividing \(-12t^3\) by \(t\): \(-12t^2\).5. Continue these steps until you obtain the quotient \(-4t^3 + 12t^2 + 36t + 119\) and remainder \(350\)...

Identify Restrictions (b)

Identify any restrictions: The denominator \(t - 3\) cannot be zero, so \(t eq 3\).

Key concepts

Long Division of Polynomials

Long division of polynomials is similar to long division with numbers. It involves breaking down a complex polynomial into simpler components. To divide polynomials using long division, follow these steps:
1. Set up the division by placing the divisor outside the division symbol and the dividend inside. For example, divide \(x^3 + 7x^2 - 3x + 4\) by \(x + 2\).
2. Divide the leading term of the dividend by the leading term of the divisor. Here, divide \(x^3\) by \(x\) to get \(x^2\).
3. Multiply the entire divisor by this result and subtract it from the dividend. This yields a new polynomial.
4. Repeat the process with the new polynomial.
By repeating these steps, you break down the dividend until you obtain a quotient and a remainder. Remember, long division simplifies complex equations step-by-step, helping you understand the structure of polynomials better.

Quotient and Remainder Theorem

The Quotient and Remainder Theorem states that for any polynomials \(P(x)\) and \(D(x)\), if you divide \(P(x)\) by \(D(x)\), you will get a quotient \(Q(x)\) and a remainder \(R\), such that:
\[ P(x) = D(x) \times Q(x) + R \]
For instance, dividing \(x^3 + 7x^2 - 3x + 4\) by \(x + 2\) results in:
\[ x^3 + 7x^2 - 3x + 4 = (x + 2) \times (x^2 + 5x - 13) + 30 \]
The quotient is \(x^2 + 5x - 13\) and the remainder is \(30\). The theorem helps us express polynomial division in a standardized way, showing the relationship between divisor, quotient, and remainder. This structure is crucial for understanding polynomial functions and their behaviors.

Restrictions on Variable

When performing polynomial division, identifying restrictions on the variable is crucial to avoid mathematical errors. Restrictions arise from the denominator of the divisor or any resulting expressions in the process. These restrictions prevent division by zero, which is undefined in mathematics.
For example, in the division \(x^3 + 7x^2 - 3x + 4 \div (x + 2)\), the denominator \(x + 2\) cannot be zero. Therefore, \(x eq -2\). This restriction ensures all steps in the division are valid and the solution is meaningful.
Similarly, in the division \((11t - 4t^4 - 7) \div (t - 3)\), the restriction is \(t eq 3\). Always checking for restrictions helps maintain the integrity of the solution and ensures the variable's values are within permissible limits.