To determine the quotient of [tex]\(x^2 + 7x + 12\)[/tex] divided by [tex]\(x + 4\)[/tex] using algebra tiles, we need to use polynomial division. We'll divide the polynomial step-by-step.
1. Set up the division:
Place [tex]\(x^2 + 7x + 12\)[/tex] in a grid format to be divided by [tex]\(x + 4\)[/tex].
2. Divide the leading term:
- The leading term of the dividend (inside the division grid) is [tex]\(x^2\)[/tex].
- The leading term of the divisor (outside the division grid) is [tex]\(x\)[/tex].
- Divide [tex]\(x^2\)[/tex] by [tex]\(x\)[/tex], which gives you [tex]\(x\)[/tex].
3. Multiply and subtract:
- Multiply the entire divisor [tex]\(x + 4\)[/tex] by the term you just found, [tex]\(x\)[/tex]. This gives [tex]\(x(x + 4) = x^2 + 4x\)[/tex].
- Subtract this result from the original polynomial:
[tex]\( (x^2 + 7x + 12) - (x^2 + 4x) = 3x + 12 \)[/tex].
- Your new simplified polynomial inside the grid is [tex]\(3x + 12\)[/tex].
4. Repeat the process:
- Now, divide [tex]\(3x\)[/tex] (the new leading term inside the division grid) by [tex]\(x\)[/tex] (the leading term of the divisor), which gives [tex]\(3\)[/tex].
- Multiply the entire divisor [tex]\(x + 4\)[/tex] by [tex]\(3\)[/tex]. This gives [tex]\(3(x + 4) = 3x + 12\)[/tex].
- Subtract this result from the current polynomial inside the grid:
[tex]\( (3x + 12) - (3x + 12) = 0 \)[/tex].
- There's no remainder, and the division process stops here.
5. The quotient:
- Combining the terms found in the steps above, the quotient is [tex]\(x + 3\)[/tex].
Final Answer:
The quotient of [tex]\(x^2 + 7x + 12\)[/tex] divided by [tex]\(x + 4\)[/tex] is [tex]\(x + 3\)[/tex].
To visually represent this process with algebra tiles:
- Represent [tex]\(x^2\)[/tex] with a large square tile.
- Use bars to represent [tex]\(x\)[/tex] terms (seven tiles to represent [tex]\(7x\)[/tex]).
- Use small unit squares to represent the constant term ([tex]\(12\)[/tex] unit tiles).
You'll arrange these tiles under the divisor [tex]\(x\)[/tex] and [tex]\(4\)[/tex] unit tiles horizontally. Start removing and matching the tiles as per the division process outlined above until you achieve the quotient tiles arranged neatly beside the divisor. This helps in visualizing the step-by-step reduction leading to the quotient [tex]\(x + 3\)[/tex].
