Rewrite the following as addition and subtraction expressions.
a) (x − 3) (x + 5). b) (x − 2) (x − 7).
c) 2 (y − 6) (y + 4). d) 3 (x − 1) (x + 2).
e) (t + 4)2. f) 3 (z − 5)2.
g) x (x − 6)2. h) s (s2 − 1)2.
i) (x + 1) (x + 2) (x + 3). j) (w − 5) (w − 2) (w + 1).
k) (x − 2)3. l) (x + 3)3.


Let's rewrite each expression as addition and subtraction:

### a) \((x - 3)(x + 5)\)
Expand the terms using the distributive property (FOIL):
\[
x(x + 5) - 3(x + 5) = x^2 + 5x - 3x - 15 = x^2 + 2x - 15
\]

### b) \((x - 2)(x - 7)\)
Expand the terms:
\[
x(x - 7) - 2(x - 7) = x^2 - 7x - 2x + 14 = x^2 - 9x + 14
\]

### c) \(2(y - 6)(y + 4)\)
First expand \((y - 6)(y + 4)\) and then multiply by 2:
\[
(y - 6)(y + 4) = y^2 + 4y - 6y - 24 = y^2 - 2y - 24
\]
Now, multiply by 2:
\[
2(y^2 - 2y - 24) = 2y^2 - 4y - 48
\]

### d) \(3(x - 1)(x + 2)\)
First expand \((x - 1)(x + 2)\) and then multiply by 3:
\[
(x - 1)(x + 2) = x^2 + 2x - x - 2 = x^2 + x - 2
\]
Now, multiply by 3:
\[
3(x^2 + x - 2) = 3x^2 + 3x - 6
\]

### e) \((t + 4)^2\)
Expand using the square of a binomial:
\[
(t + 4)(t + 4) = t^2 + 4t + 4t + 16 = t^2 + 8t + 16
\]

### f) \(3(z - 5)^2\)
First expand \((z - 5)^2\):
\[
(z - 5)(z - 5) = z^2 - 5z - 5z + 25 = z^2 - 10z + 25
\]
Now, multiply by 3:
\[
3(z^2 - 10z + 25) = 3z^2 - 30z + 75
\]

### g) \(x(x - 6)^2\)
First expand \((x - 6)^2\):
\[
(x - 6)(x - 6) = x^2 - 6x - 6x + 36 = x^2 - 12x + 36
\]
Now multiply by \(x\):
\[
x(x^2 - 12x + 36) = x^3 - 12x^2 + 36x
\]

### h) \(s(s^2 - 1)^2\)
First expand \((s^2 - 1)^2\):
\[
(s^2 - 1)(s^2 - 1) = s^4 - s^2 - s^2 + 1 = s^4 - 2s^2 + 1
\]
Now multiply by \(s\):
\[
s(s^4 - 2s^2 + 1) = s^5 - 2s^3 + s
\]

### i) \((x + 1)(x + 2)(x + 3)\)
Expand step by step:
\[
(x + 1)(x + 2) = x^2 + 2x + x + 2 = x^2 + 3x + 2
\]
Now multiply by \((x + 3)\):
\[
(x^2 + 3x + 2)(x + 3) = x^3 + 3x^2 + 2x + 3x^2 + 9x + 6 = x^3 + 6x^2 + 11x + 6
\]

### j) \((w - 5)(w - 2)(w + 1)\)
Expand step by step:
\[
(w - 5)(w - 2) = w^2 - 2w - 5w + 10 = w^2 - 7w + 10
\]
Now multiply by \((w + 1)\):
\[
(w^2 - 7w + 10)(w + 1) = w^3 + w^2 - 7w^2 - 7w + 10w + 10 = w^3 - 6w^2 + 3w + 10
\]

### k) \((x - 2)^3\)
Expand using the cube of a binomial:
\[
(x - 2)^3 = (x - 2)(x - 2)(x - 2) = (x^2 - 4x + 4)(x - 2) = x^3 - 2x^2 - 4x^2 + 8x + 4x - 8 = x^3 - 6x^2 + 12x - 8
\]

### l) \((x + 3)^3\)
Expand using the cube of a binomial:
\[
(x + 3)^3 = (x + 3)(x + 3)(x + 3) = (x^2 + 6x + 9)(x + 3) = x^3 + 3x^2 + 6x^2 + 18x + 9x + 27 = x^3 + 9x^2 + 27x + 27
\]

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Do you want more details or have any questions?

Here are 5 related questions to consider:
1. How can we expand and simplify \((x + a)(x + b)\)?
2. What is the general formula for expanding \((x - a)^2\)?
3. How do we expand expressions like \((x + a)^3\)?
4. What is the shortcut for expanding binomials raised to a power?
5. How can we apply the distributive property to multiply three binomials?

**Tip:** Always check if you can simplify expressions before expanding them—it saves time and reduces mistakes!

Rewrite the following as addition and subtraction expressions.
 a) (x − 3) (x + 5). b) (x − 2) (x − 7).
 c) 2 (y − 6) (y + 4). d) 3 (x − 1) (x + 2).
 e) (t + 4)^2. f) 3 (z − 5)^2.
 g) x (x − 6)^2. h) s (s^2 − 1)^2.
 i) (x + 1) (x + 2) (x + 3). j) (w − 5) (w − 2) (w + 1).
 k) (x − 2)^3. l) (x + 3)^3.

Learn how to expand and simplify algebraic expressions like (x − 3)(x + 5) and (x + 1)(x + 2)(x + 3) using polynomial expansion techniques. This problem set covers key concepts in algebra, including the distributive property and binomial/cubic expansions. Ideal for high school students in Grades 9-12.

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