Question

1. factor each expression, if possible. then, check with distribution.

a) y² - 49
b) 25 - 16w⁴
c) 18 - 8w⁴
d) 36ax - a⁵x⁵
e) x² + 9

1. similar to one of the problems in wiley: factor (a + b)² - 100
2. construct an expression with two terms that can be factored, and factor it.
3. construct an expression with two terms that cannot be factored, and demonstrate why it is prime.

Explanation:

Step1: Recall the difference of squares formula

The difference of squares formula is $a^{2}-b^{2}=(a + b)(a - b)$.

Step2: Factor $y^{2}-49$

Here $a = y$ and $b = 7$, so $y^{2}-49=y^{2}-7^{2}=(y + 7)(y - 7)$.

Step3: Factor $25-16w^{4}$

Rewrite it as $5^{2}-(4w^{2})^{2}$. Using the difference of squares formula with $a = 5$ and $b = 4w^{2}$, we get $(5 + 4w^{2})(5 - 4w^{2})$.

Step4: Factor $18 - 8w^{4}$

First, factor out the greatest common factor 2: $2(9 - 4w^{4})$. Then rewrite $9 - 4w^{4}$ as $3^{2}-(2w^{2})^{2}$. Using the difference of squares formula, we have $2(3 + 2w^{2})(3 - 2w^{2})$.

Step5: Factor $36ax - a^{5}x^{5}$

Factor out the greatest common factor $ax$: $ax(36 - a^{4}x^{4})$. Rewrite $36 - a^{4}x^{4}$ as $6^{2}-(a^{2}x^{2})^{2}$. Then factor it as $ax(6 + a^{2}x^{2})(6 - a^{2}x^{2})$.

Step6: Analyze $x^{2}+9$

For a quadratic of the form $x^{2}+bx + c$ to be factored as $(x + m)(x + n)=x^{2}+(m + n)x+mn$, for $x^{2}+9=x^{2}+0x + 9$, there are no real - numbers $m$ and $n$ such that $mn = 9$ and $m + n = 0$.

Step7: Factor $(a + b)^{2}-100$

Rewrite it as $(a + b)^{2}-10^{2}$. Using the difference of squares formula with $a=a + b$ and $b = 10$, we get $(a + b+10)(a + b - 10)$.

Step8: Construct and factor an expression

For example, for $x^{2}-16$, we have $a=x$ and $b = 4$ in the difference of squares formula, so $x^{2}-16=(x + 4)(x - 4)$.

Step9: Construct a non - factorable expression

For $x^{2}+1$, assume it can be factored as $(x + m)(x + n)=x^{2}+(m + n)x+mn$. Then $mn = 1$ and $m + n = 0$. Solving $m + n = 0$ gives $n=-m$, and substituting into $mn = 1$ gives $-m^{2}=1$ or $m^{2}=-1$, which has no real - number solutions.

Answer:

a) $(y + 7)(y - 7)$
b) $(5 + 4w^{2})(5 - 4w^{2})$
c) $2(3 + 2w^{2})(3 - 2w^{2})$
d) $ax(6 + a^{2}x^{2})(6 - a^{2}x^{2})$
e) Cannot be factored over the real - numbers

1. $(a + b+10)(a + b - 10)$
2. Example: $x^{2}-16=(x + 4)(x - 4)$
3. Example: $x^{2}+1$. There are no two real - numbers $m$ and $n$ such that $mn = 1$ and $m + n=0$.