QUESTION IMAGE
Question
- factor completely each of the following expressions.
(a) \\(12 + x - x^2\\)
(b) \\(2p^3 - 12p^2 + 16p\\)
(c) \\(8x^2 - 10x - 3\\)
(d) \\(63x^2 - 33x - 6\\)
(e) \\(49y^2 - 81\\)
(f) \\(25p^2 - 30p + 9\\)
(g) \\(10a^2 + 11ab - 6b^2\\)
(h) \\(200x^2 - 18\\)
(i) \\(16u^2 + 20u - 6\\)
(j) \\(w^2 - 3w - 28\\)
(k) \\(-18x^3 - 15x^2 + 18x\\)
(l) \\(3y^2 - 9y - 30\\)
<pre_analysis>
{
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"question_count": 12,
"skills_matched": [
"step_cot",
"table_formatter",
"explain_then_answer",
"tutor"
],
"needs_drawing": false,
"concepts_used": [
"Polynomial Factoring",
"Difference of Squares"
],
"new_concepts": [
"Quadratic Factoring",
"Greatest Common Factor"
],
"current_concepts": [
"Polynomial Factoring",
"Difference of Squares",
"Quadratic Factoring",
"Greatest Common Factor"
]
}
</pre_analysis>
<reasoning>
Factor expressions (a) to (c)
Using the Polynomial Factoring knowledge point, we factor the first three quadratic expressions.
For (a):
\[
12 + x - x^2 = -(x^2 - x - 12) = -(x - 4)(x + 3) = (4 - x)(3 + x)
\]
For (b), we first extract the greatest common factor \(2p\):
\[
2p^3 - 12p^2 + 16p = 2p(p^2 - 6p + 8) = 2p(p - 4)(p - 2)
\]
For (c), we use splitting the middle term:
\[
8x^2 - 10x - 3 = 8x^2 - 12x + 2x - 3 = 4x(2x - 3) + 1(2x - 3) = (4x + 1)(2x - 3)
\]
Factor expressions (d) to (f)
Using the Polynomial Factoring and Difference of Squares knowledge points:
For (d), we factor out the common factor \(3\):
\[
63x^2 - 33x - 6 = 3(21x^2 - 11x - 2) = 3(21x^2 - 14x + 3x - 2) = 3[7x(3x - 2) + 1(3x - 2)] = 3(7x + 1)(3x - 2)
\]
For (e), we apply the difference of squares:
\[
49y^2 - 81 = (7y)^2 - 9^2 = (7y - 9)(7y + 9)
\]
For (f), we recognize a perfect square trinomial:
\[
25p^2 - 30p + 9 = (5p)^2 - 2(5p)(3) + 3^2 = (5p - 3)^2
\]
Factor expressions (g) to (i)
Using the Polynomial Factoring and Difference of Squares knowledge points:
For (g), we factor the two-variable quadratic:
\[
10a^2 + 11ab - 6b^2 = 10a^2 + 15ab - 4ab - 6b^2 = 5a(2a + 3b) - 2b(2a + 3b) = (5a - 2b)(2a + 3b)
\]
For (h), we factor out the common factor \(2\):
\[
200x^2 - 18 = 2(100x^2 - 9) = 2[(10x)^2 - 3^2] = 2(10x - 3)(10x + 3)
\]
For (i), we factor out the common factor \(2\):
\[
16u^2 + 20u - 6 = 2(8u^2 + 10u - 3) = 2(8u^2 + 12u - 2u - 3) = 2[4u(2u + 3) - 1(2u + 3)] = 2(4u - 1)(2u + 3)
\]
Factor expressions (j) to (l)
Using the Polynomial Factoring knowledge point:
For (j):
\[
w^2 - 3w - 28 = (w - 7)(w + 4)
\]
For (k), we factor out the common factor \(-3x\):
\[
-18x^3 - 15x^2 + 18x = -3x(6x^2 + 5x - 6) = -3x(6x^2 + 9x - 4x - 6) = -3x[3x(2x + 3) - 2(2x + 3)] = -3x(3x - 2)(2x + 3)
\]
For (l), we factor out the common factor \(3\):
\[
3y^2 - 9y - 30 = 3(y^2 - 3y - 10) = 3(y - 5)(y + 2)
\]
</reasoning>
<answer>
| No. | Problem | Answer |
|---|---|---|
| (b) | \(2p^3 - 12p^2 + 16p\) | \(2p(p - 4)(p - 2)\) |
| (c) | \(8x^2 - 10x - 3\) | \((4x + 1)(2x - 3)\) |
| (d) | \(63x^2 - 33x - 6\) | \(3(7x + 1)(3x - 2)\) |
| (e) | \(49y^2 - 81\) | \((7y - 9)(7y + 9)\) |
| (f) | \(25p^2 - 30p + 9\) | \((5p - 3)^2\) |
| (g) | \(10a^2 + 11ab - 6b^2\) | \((5a - 2b)(2a + 3b)\) |
| (h) | \(200x^2 - 18\) | \(2(10x - 3)(10x + 3)\) |
| (i) | \(16u^2 + 20u - 6\) | \(2(4u - 1)(2u + 3)\) |
| (j) | \(w^2 - 3w - 28\) | \((w - 7)(w + 4)\) |
| (k) | \(-18x^3 - 15x^2 + 18x\) | \(-3x(3x - 2)(2x + 3)\) |
| (l) | \(3y^2 - 9y - 30\) | \(3(y - 5)(y + 2)\) |
</answer>
<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Algebra",
"Polynomial Factoring"
]
}
</post_analysis>
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<pre_analysis>
{
"quality": "clear",
"question_count": 12,
"skills_matched": [
"step_cot",
"table_formatter",
"explain_then_answer",
"tutor"
],
"needs_drawing": false,
"concepts_used": [
"Polynomial Factoring",
"Difference of Squares"
],
"new_concepts": [
"Quadratic Factoring",
"Greatest Common Factor"
],
"current_concepts": [
"Polynomial Factoring",
"Difference of Squares",
"Quadratic Factoring",
"Greatest Common Factor"
]
}
</pre_analysis>
<reasoning>
Factor expressions (a) to (c)
Using the Polynomial Factoring knowledge point, we factor the first three quadratic expressions.
For (a):
\[
12 + x - x^2 = -(x^2 - x - 12) = -(x - 4)(x + 3) = (4 - x)(3 + x)
\]
For (b), we first extract the greatest common factor \(2p\):
\[
2p^3 - 12p^2 + 16p = 2p(p^2 - 6p + 8) = 2p(p - 4)(p - 2)
\]
For (c), we use splitting the middle term:
\[
8x^2 - 10x - 3 = 8x^2 - 12x + 2x - 3 = 4x(2x - 3) + 1(2x - 3) = (4x + 1)(2x - 3)
\]
Factor expressions (d) to (f)
Using the Polynomial Factoring and Difference of Squares knowledge points:
For (d), we factor out the common factor \(3\):
\[
63x^2 - 33x - 6 = 3(21x^2 - 11x - 2) = 3(21x^2 - 14x + 3x - 2) = 3[7x(3x - 2) + 1(3x - 2)] = 3(7x + 1)(3x - 2)
\]
For (e), we apply the difference of squares:
\[
49y^2 - 81 = (7y)^2 - 9^2 = (7y - 9)(7y + 9)
\]
For (f), we recognize a perfect square trinomial:
\[
25p^2 - 30p + 9 = (5p)^2 - 2(5p)(3) + 3^2 = (5p - 3)^2
\]
Factor expressions (g) to (i)
Using the Polynomial Factoring and Difference of Squares knowledge points:
For (g), we factor the two-variable quadratic:
\[
10a^2 + 11ab - 6b^2 = 10a^2 + 15ab - 4ab - 6b^2 = 5a(2a + 3b) - 2b(2a + 3b) = (5a - 2b)(2a + 3b)
\]
For (h), we factor out the common factor \(2\):
\[
200x^2 - 18 = 2(100x^2 - 9) = 2[(10x)^2 - 3^2] = 2(10x - 3)(10x + 3)
\]
For (i), we factor out the common factor \(2\):
\[
16u^2 + 20u - 6 = 2(8u^2 + 10u - 3) = 2(8u^2 + 12u - 2u - 3) = 2[4u(2u + 3) - 1(2u + 3)] = 2(4u - 1)(2u + 3)
\]
Factor expressions (j) to (l)
Using the Polynomial Factoring knowledge point:
For (j):
\[
w^2 - 3w - 28 = (w - 7)(w + 4)
\]
For (k), we factor out the common factor \(-3x\):
\[
-18x^3 - 15x^2 + 18x = -3x(6x^2 + 5x - 6) = -3x(6x^2 + 9x - 4x - 6) = -3x[3x(2x + 3) - 2(2x + 3)] = -3x(3x - 2)(2x + 3)
\]
For (l), we factor out the common factor \(3\):
\[
3y^2 - 9y - 30 = 3(y^2 - 3y - 10) = 3(y - 5)(y + 2)
\]
</reasoning>
<answer>
| No. | Problem | Answer |
|---|---|---|
| (b) | \(2p^3 - 12p^2 + 16p\) | \(2p(p - 4)(p - 2)\) |
| (c) | \(8x^2 - 10x - 3\) | \((4x + 1)(2x - 3)\) |
| (d) | \(63x^2 - 33x - 6\) | \(3(7x + 1)(3x - 2)\) |
| (e) | \(49y^2 - 81\) | \((7y - 9)(7y + 9)\) |
| (f) | \(25p^2 - 30p + 9\) | \((5p - 3)^2\) |
| (g) | \(10a^2 + 11ab - 6b^2\) | \((5a - 2b)(2a + 3b)\) |
| (h) | \(200x^2 - 18\) | \(2(10x - 3)(10x + 3)\) |
| (i) | \(16u^2 + 20u - 6\) | \(2(4u - 1)(2u + 3)\) |
| (j) | \(w^2 - 3w - 28\) | \((w - 7)(w + 4)\) |
| (k) | \(-18x^3 - 15x^2 + 18x\) | \(-3x(3x - 2)(2x + 3)\) |
| (l) | \(3y^2 - 9y - 30\) | \(3(y - 5)(y + 2)\) |
</answer>
<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Algebra",
"Polynomial Factoring"
]
}
</post_analysis>