Question

a) Use long division to divide \(x^{2}+10 x-24\) by \(x-2 .\) Express the result in the form \(\frac{P(x)}{x-a}=Q(x)+\frac{R}{x-a}\) b) Identify any restrictions on the variable. c) Write the corresponding statement that can be used to check the division. d) Verify your answer.

Short answer

1. \frac{x^2 + 10x - 24}{x-2} = x + 12\ 2. \(x eq 2\) \ 3. \ (x + 12)(x - 2) + 0 = x^2 + 10x - 24\.

Step-by-step solution

- Set up the long division

Write down the polynomial division \(x^2 + 10x - 24\) divided by \(x-2\). Place \(x^2 + 10x - 24\) under the long division symbol and \(x-2\) outside.

- Perform the division

Determine how many times \(x\) goes into \(x^2\). Write \(x\) above the long division symbol. Multiply \(x-2\) by \(x\) to get \(x^2 - 2x\) and subtract this from \(x^2 + 10x - 24\). The result is \(12x - 24\).

- Complete the division

Next, determine how many times \(x\) goes into \(12x\). Write \(+12\) above the long division symbol. Multiply \(x-2\) by 12 to get \(12x - 24\) and subtract this from \(12x - 24\). The remainder is 0.

- Write the division result

Since the remainder is 0, the result of the division is \x + 12\. So, in the form \frac{P(x)}{x-a}=Q(x)+\frac{R}{x-a},\ we have \[\frac{x^2 + 10x - 24}{x-2} = x + 12 + \frac{0}{x-2} \].

- Identify restrictions on the variable

The denominator \(x-2\) cannot be zero, so \(x\) cannot be 2. Therefore, the restriction on the variable is \(x eq 2\).

- Write the statement to check the division

To check the division, multiply the quotient by the divisor and add the remainder: \[ (x + 12)(x - 2) + 0 = x^2 + 10x - 24 \].

- Verify the answer

Expand \( (x + 12)(x - 2) = x(x - 2) + 12(x - 2) = x^2 - 2x + 12x - 24 = x^2 + 10x - 24 \). This confirms that the division was performed correctly.

Key concepts

Long Division

Polynomial long division is a method used to divide polynomials, similar to the way we divide numbers. In our example, we divide the polynomial \( x^2 + 10x - 24 \) by \( x - 2 \).

Here's a step-by-step breakdown to make it easier:

• First, set up the division with \( x^2 + 10x - 24 \) under the division symbol and \( x - 2 \) outside.
• Determine how many times the leading term of the divisor (\( x \)) can go into the leading term of the dividend (\( x^2 \)) — that's \( x \times x = x^2 \). Write \( x \) above the division symbol.
• Multiply \( x-2 \) by \( x \) resulting in \( x^2 - 2x \), then subtract this from \( x^2 + 10x - 24 \).
• Now, bring down the next term (\( 12x - 24 \)). Repeat the process by dividing \( 12x \) by \( x \), which gives \( 12 \). Write \( +12 \) next to \( x \) above the division symbol.
• Multiply and subtract again, which leaves a remainder of zero.

The quotient is \( x + 12 \) and the remainder is zero.

Quotient and Remainder

The division process yields two important components: the quotient and the remainder.

In this particular example:

• The quotient is \( Q(x) = x + 12 \).
• The remainder is \( R = 0 \).

We express the result of our polynomial division in the form: \[ \frac{P(x)}{x-a} = Q(x) + \frac{R}{x-a} \] Given the result of our division, we have: \[ \frac{x^2+10x-24}{x-2} = x+12+\frac{0}{x-2} \] This simplified representation allows for easier understanding and manipulation of polynomial expressions.

Polynomial Restrictions

Polynomial division is generally straightforward, but we must consider certain restrictions on the variable due to the denominator.

In our case, the divisor is \( x-2 \).

• The denominator cannot be zero, so we set \( x - 2 eq 0 \).
• Solving this inequality, we find \( x eq 2 \).

Thus, the restriction on the variable is that \( x \) cannot be 2. Ignoring this restriction could lead to undefined expressions or mathematical errors in subsequent operations.

Verification of Division

Verification of polynomial division ensures that our solution is accurate. We do this by multiplying the quotient by the divisor and adding the remainder.

For the given problem:

• The quotient is \( x + 12 \).
• The divisor is \( x - 2 \).
• The remainder is 0.

We follow these steps:

1. Multiply the quotient and the divisor: \t\[ (x + 12)(x - 2) \ x(x - 2) + 12(x - 2) \ = x^2 - 2x + 12x - 24 \ = x^2 + 10x - 24 \] 2. Add the remainder (0 in this case) to the result of our multiplication: \[ x^2 + 10x - 24 + 0 = x^2 + 10x - 24 \] This confirms that our polynomial division was conducted correctly and verifies that \( x^2 + 10x - 24 = (x+12)(x-2) \).