Question

1. (5p - 4)^2
2. (8 - 7m)(8 + 7m)
3. (7x + 5)^2
4. (6p + 5)(6p^2+7p - 1)
5. (6p + 7)(3p^2 + 6p + 1)
6. (3x - 2)(8x^2 - 7x + 7)

factor each completely.

1. p^2 - 7p
2. n^2 + 4n - 21
3. k^2 + 7k - 30
4. x^2 + 5x - 6
5. n^2 - 17n + 72
6. x^2 - 13x + 36
7. 2m^2 - 22m + 20
8. 3n^2 - 3n - 168
9. 3x^2 + 24x - 60

Explanation:

Step1: Expand $(5p - 4)^2$

Use the formula $(a - b)^2=a^{2}-2ab + b^{2}$, where $a = 5p$ and $b = 4$. So, $(5p-4)^2=(5p)^{2}-2\times(5p)\times4 + 4^{2}=25p^{2}-40p + 16$.

Step2: Expand $(8 - 7m)(8 + 7m)$

Use the difference - of - squares formula $(a - b)(a + b)=a^{2}-b^{2}$, where $a = 8$ and $b = 7m$. So, $(8 - 7m)(8 + 7m)=8^{2}-(7m)^{2}=64 - 49m^{2}$.

Step3: Expand $(7x + 5)^2$

Use the formula $(a + b)^2=a^{2}+2ab + b^{2}$, where $a = 7x$ and $b = 5$. So, $(7x + 5)^2=(7x)^{2}+2\times(7x)\times5+5^{2}=49x^{2}+70x + 25$.

Step4: Expand $(6p + 5)(6p^{2}+7p - 1)$

Use the distributive property:
\[

$$\begin{align*} &(6p + 5)(6p^{2}+7p - 1)\\ =&6p\times(6p^{2}+7p - 1)+5\times(6p^{2}+7p - 1)\\ =&36p^{3}+42p^{2}-6p + 30p^{2}+35p - 5\\ =&36p^{3}+(42p^{2}+30p^{2})+( - 6p+35p)-5\\ =&36p^{3}+72p^{2}+29p - 5 \end{align*}$$

\]

Step5: Expand $(6p + 7)(3p^{2}+6p + 1)$

Use the distributive property:
\[

$$\begin{align*} &(6p + 7)(3p^{2}+6p + 1)\\ =&6p\times(3p^{2}+6p + 1)+7\times(3p^{2}+6p + 1)\\ =&18p^{3}+36p^{2}+6p+21p^{2}+42p + 7\\ =&18p^{3}+(36p^{2}+21p^{2})+(6p + 42p)+7\\ =&18p^{3}+57p^{2}+48p + 7 \end{align*}$$

\]

Step6: Expand $(3x - 2)(8x^{2}-7x + 7)$

Use the distributive property:
\[

$$\begin{align*} &(3x - 2)(8x^{2}-7x + 7)\\ =&3x\times(8x^{2}-7x + 7)-2\times(8x^{2}-7x + 7)\\ =&24x^{3}-21x^{2}+21x-16x^{2}+14x - 14\\ =&24x^{3}+(-21x^{2}-16x^{2})+(21x + 14x)-14\\ =&24x^{3}-37x^{2}+35x - 14 \end{align*}$$

\]

Step7: Factor $p^{2}-7p$

Factor out the common factor $p$: $p^{2}-7p=p(p - 7)$.

Step8: Factor $n^{2}+4n - 21$

Find two numbers that multiply to $-21$ and add up to $4$. The numbers are $7$ and $-3$. So, $n^{2}+4n - 21=(n + 7)(n - 3)$.

Step9: Factor $k^{2}+7k - 30$

Find two numbers that multiply to $-30$ and add up to $7$. The numbers are $10$ and $-3$. So, $k^{2}+7k - 30=(k + 10)(k - 3)$.

Step10: Factor $x^{2}+5x - 6$

Find two numbers that multiply to $-6$ and add up to $5$. The numbers are $6$ and $-1$. So, $x^{2}+5x - 6=(x + 6)(x - 1)$.

Step11: Factor $n^{2}-17n + 72$

Find two numbers that multiply to $72$ and add up to $-17$. The numbers are $-8$ and $-9$. So, $n^{2}-17n + 72=(n - 8)(n - 9)$.

Step12: Factor $x^{2}-13x + 36$

Find two numbers that multiply to $36$ and add up to $-13$. The numbers are $-4$ and $-9$. So, $x^{2}-13x + 36=(x - 4)(x - 9)$.

Step13: Factor $2m^{2}-22m + 20$

First, factor out the common factor $2$: $2m^{2}-22m + 20=2(m^{2}-11m + 10)$. Then, find two numbers that multiply to $10$ and add up to $-11$. The numbers are $-10$ and $-1$. So, $2m^{2}-22m + 20=2(m - 10)(m - 1)$.

Step14: Factor $3n^{2}-3n - 168$

First, factor out the common factor $3$: $3n^{2}-3n - 168=3(n^{2}-n - 56)$. Then, find two numbers that multiply to $-56$ and add up to $-1$. The numbers are $-8$ and $7$. So, $3n^{2}-3n - 168=3(n - 8)(n+7)$.

Step15: Factor $3x^{2}+24x - 60$

First, factor out the common factor $3$: $3x^{2}+24x - 60=3(x^{2}+8x - 20)$. Then, find two numbers that multiply to $-20$ and add up to $8$. The numbers are $10$ and $-2$. So, $3x^{2}+24x - 60=3(x + 10)(x - 2)$.

Answer:

1. $25p^{2}-40p + 16$
2. $64 - 49m^{2}$
3. $49x^{2}+70x + 25$
4. $36p^{3}+72p^{2}+29p - 5$
5. $18p^{3}+57p^{2}+48p + 7$
6. $24x^{3}-37x^{2}+35x - 14$
7. $p(p - 7)$
8. $(n + 7)(n - 3)$
9. $(k + 10)(k - 3)$
10. $(x + 6)(x - 1)$
11. $(n - 8)(n - 9)$
12. $(x - 4)(x - 9)$
13. $2(m - 10)(m - 1)$
14. $3(n - 8)(n + 7)$
15. $3(x + 10)(x - 2)$