Question

name:
group:
factoring by greatest common factor
factor the greatest common factor out of the polynomial

1. $8x^3 + 10x$
2. $12y - 16$
3. $-15d^8 + 45d^9$
4. $13a + 20b$
5. $c^4 + c^3 - c$
6. $6n^2 - 30n + 42$
7. $-7m^2 - 10m + 17$
8. $18p^3 - 63p^2 - 9p$
9. $18x^3 - 50y^3$
10. $100z^8 + 50z^6 - 75z^4$
11. $36rs^3 - 108r^3s^3$
12. $36k - 30$
13. $a^7b - a^{10}$
14. $2c^3d^4 - 3c^4 + 4c^3$
15. $3g^8 + 3g^7$
16. $18x^3 - 48x^4 + 56x^3 - 86x$
17. $23y^{10} - 46y^7 + 68y^3 + 10y$

Explanation:

Let's solve problem 1: \(8x^{3}+10x\) using the step - by - step format.

Step 1: Find the GCF of coefficients and variables

For the coefficients 8 and 10, we factor them: \(8 = 2\times4\) and \(10=2\times5\). So the greatest common factor (GCF) of 8 and 10 is 2.
For the variable terms \(x^{3}\) and \(x\), using the rule of exponents \(x^{m}\div x^{n}=x^{m - n}\) (\(m\geq n\)), the GCF of \(x^{3}\) and \(x\) is \(x\) (since \(x^{3}\div x=x^{2}\) and \(x\div x = 1\)).
So the GCF of \(8x^{3}\) and \(10x\) is \(2x\).

Step 2: Factor out the GCF

We divide each term in the polynomial \(8x^{3}+10x\) by the GCF \(2x\):
\(\frac{8x^{3}}{2x}+\frac{10x}{2x}=4x^{2} + 5\)
So, \(8x^{3}+10x=2x(4x^{2}+5)\)

Answer:

\(2x(4x^{2}+5)\)