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Magazine Chapter 6: Problem 22
Divide and check. $$ \left(a^{2}-8 a-16\right) \div(a+4) $$
Short Answer
Expert verified
The quotient is \(a - 12\) with a remainder of \(32\).
Step by step solution
01
Set Up Polynomial Long Division
Rewrite the problem in long division form. The dividend is the polynomial \(a^2 - 8a - 16\) and the divisor is \(a + 4\).
02
Divide the Leading Terms
Divide the leading term of the dividend \(a^2\) by the leading term of the divisor \(a\). This gives \(a\).
03
Multiply and Subtract
Multiply \(a\) by \(a + 4\) to get \(a^2 + 4a\). Subtract this from \(a^2 - 8a - 16\), which results in \(-12a - 16\).
04
Divide the Next Term
Divide the new leading term \(-12a\) by \(a\). This gives \(-12\).
05
Multiply and Subtract Again
Multiply \(-12\) by \(a + 4\) to get \(-12a - 48\). Subtract this from \(-12a - 16\), resulting in \(32\).
06
Write the Quotient and Remainder
The quotient is \(a - 12\), and the remainder is \(32\). So, \( (a^2 - 8a - 16) \div (a + 4) = a - 12\) with a remainder of \(32\).
07
Check the Solution
To check, multiply the quotient \(a - 12\) by the divisor \(a + 4\) and add the remainder:\( (a - 12)(a + 4) + 32 = a^2 - 8a - 16\). This confirms the correctness of the solution.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Long Division
Polynomial long division works similarly to numerical long division. It's used to divide one polynomial by another, resulting in a quotient and sometimes a remainder.
To start, we rewrite the division problem in long division form. Here, we're dividing the polynomial \(a^2 - 8a - 16\) by \(a + 4\).
We first focus on the leading terms. Divide the leading term of the dividend \(a^2\) by the leading term of the divisor \(a\). This results in \(a\).
Next, multiply this quotient term by the entire divisor and subtract it from the original polynomial like this:
To start, we rewrite the division problem in long division form. Here, we're dividing the polynomial \(a^2 - 8a - 16\) by \(a + 4\).
We first focus on the leading terms. Divide the leading term of the dividend \(a^2\) by the leading term of the divisor \(a\). This results in \(a\).
Next, multiply this quotient term by the entire divisor and subtract it from the original polynomial like this:
- Multiply: \(a \times (a + 4) = a^2 + 4a\).
- Subtract: \(a^2 - 8a - 16 - (a^2 + 4a)\) leads us to \(-12a - 16\).
Polynomial Remainder
The polynomial remainder is what鈥檚 left after performing the polynomial division.
In our example, once we completed dividing and subtracting the leading terms, we reached a point where further division by the leading term of the divisor wasn鈥檛 feasible.
For the second round of division, we divide \(-12a\) by \(a\), resulting in \(-12\).
In our example, once we completed dividing and subtracting the leading terms, we reached a point where further division by the leading term of the divisor wasn鈥檛 feasible.
For the second round of division, we divide \(-12a\) by \(a\), resulting in \(-12\).
- Multiply: \(-12 \times (a + 4) = -12a - 48\)
- Subtract: \(-12a - 16 - (-12a - 48) = 32\)
Checking Algebraic Solutions
Checking your solution ensures correctness and helps build confidence in your method.
To check the solution, multiply the quotient by the divisor and add the remainder back.
For our division problem, we have:
\((a - 12)(a + 4) + 32\).
Expanding this, we get:
To check the solution, multiply the quotient by the divisor and add the remainder back.
For our division problem, we have:
- Quotient: \(a - 12\)
- Divisor: \(a + 4\)
- Remainder: \(32\)
\((a - 12)(a + 4) + 32\).
Expanding this, we get:
- Expand: \(a(a+4) - 12(a+4)\)
- Sum up: \(a^2 + 4a - 12a - 48 + 32\)
- Simplify: \(a^2 - 8a - 16\)
Algebraic Operations
Understanding and performing algebraic operations accurately is crucial in polynomial division.
These operations include multiplication, division, addition, and subtraction of polynomial terms.
These operations include multiplication, division, addition, and subtraction of polynomial terms.
- Multiplication involves distributing each term of one polynomial to every term of another.
- Division, especially polynomial long division, involves repeatedly dividing the leading terms and then subtracting the products.
- Addition and subtraction of polynomials involve combining like terms to simplify expressions.
