Question

leveled practice in 1 - 4, factor each algebraic expression.

1. 12x² - 9x

=3·□·x·x - 3·3·□
=□x(4x - □)

1. - 10m³ + 25m²

=(-1)·5·□·m·m·□
+5·5·□·m
=□m□(□m + □)

1. 24n⁵ - 16n³

=□n□(3n□ - 2)

1. - 21a⁶b⁸ + 35a²b⁹

=□a□b□(-3a⁴ + 5b)

1. factor 63k⁴ - 27k³.
2. analyze and persevere use factoring to write two different equivalent expressions for - 18d² - 24d.
3. completely factor 48x⁷y⁴ + 16x³y⁵.
4. check for reasonableness jia says she completely factored - 14t⁹ + 42t⁶ as 7t³(-2t⁶ + 6t³). is she correct? explain.
5. apply math models a landscaper is planning a rectangle - shaped flower garden with an area given by the expression 4p² + 12p square yards. draw one possible design for the flower garden and label the dimensions for the length and width.

Explanation:

Step1: Factor out GCF for \(12x^{2}-9x\)

Find the greatest - common factor of \(12x^{2}\) and \(9x\). The GCF of 12 and 9 is 3, and the GCF of \(x^{2}\) and \(x\) is \(x\). So \(12x^{2}-9x = 3\cdot4\cdot x\cdot x-3\cdot3\cdot x=3x(4x - 3)\).

Step2: Factor out GCF for \(-10m^{3}+25m^{2}\)

The GCF of 10 and 25 is 5, and the GCF of \(m^{3}\) and \(m^{2}\) is \(m^{2}\). So \(-10m^{3}+25m^{2}=(-1)\cdot5\cdot2\cdot m\cdot m\cdot m + 5\cdot5\cdot m\cdot m=5m^{2}(-2m + 5)\).

Step3: Factor out GCF for \(24n^{5}-16n^{3}\)

The GCF of 24 and 16 is 8, and the GCF of \(n^{5}\) and \(n^{3}\) is \(n^{3}\). So \(24n^{5}-16n^{3}=8n^{3}(3n^{2}-2)\).

Step4: Factor out GCF for \(-21a^{6}b^{8}+35a^{2}b^{9}\)

The GCF of 21 and 35 is 7, the GCF of \(a^{6}\) and \(a^{2}\) is \(a^{2}\), and the GCF of \(b^{8}\) and \(b^{9}\) is \(b^{8}\). So \(-21a^{6}b^{8}+35a^{2}b^{9}=7a^{2}b^{8}(-3a^{4}+5b)\).

Step5: Factor out GCF for \(63k^{4}-27k^{3}\)

The GCF of 63 and 27 is 9, and the GCF of \(k^{4}\) and \(k^{3}\) is \(k^{3}\). So \(63k^{4}-27k^{3}=9k^{3}(7k - 3)\).

Step6: Factor \(-18d^{2}-24d\)

The GCF of 18 and 24 is 6, and the GCF of \(d^{2}\) and \(d\) is \(d\). So \(-18d^{2}-24d=-6d(3d + 4)\) and also \(18d^{2}+24d = 6d(3d + 4)\), then \(-18d^{2}-24d=- 2\cdot3d(3d + 4)\).

Step7: Factor \(48x^{7}y^{4}+16x^{3}y^{5}\)

The GCF of 48 and 16 is 16, the GCF of \(x^{7}\) and \(x^{3}\) is \(x^{3}\), and the GCF of \(y^{4}\) and \(y^{5}\) is \(y^{4}\). So \(48x^{7}y^{4}+16x^{3}y^{5}=16x^{3}y^{4}(3x^{4}+y)\).

Step8: Check \( - 14t^{9}+42t^{6}\)

The GCF of 14 and 42 is 14, and the GCF of \(t^{9}\) and \(t^{6}\) is \(t^{6}\). So \(-14t^{9}+42t^{6}=14t^{6}(-t^{3}+3)\), Jia is not correct.

Step9: Factor \(4p^{2}+12p\)

The GCF of 4 and 12 is 4, and the GCF of \(p^{2}\) and \(p\) is \(p\). So \(4p^{2}+12p = 4p(p + 3)\). One possible design for the rectangle - shaped flower garden: length \(4p\) yards and width \((p + 3)\) yards.

Answer:

1. \(3x(4x - 3)\)
2. \(5m^{2}(-2m + 5)\)
3. \(8n^{3}(3n^{2}-2)\)
4. \(7a^{2}b^{8}(-3a^{4}+5b)\)
5. \(9k^{3}(7k - 3)\)
6. \(-6d(3d + 4)\) and \(-2\cdot3d(3d + 4)\)
7. \(16x^{3}y^{4}(3x^{4}+y)\)
8. No, the correct factorization is \(14t^{6}(-t^{3}+3)\)
9. Length \(4p\) yards, width \((p + 3)\) yards