Question

3.1 - 3.2: factor the following polynomials

1. (3p^{2}+9p - 210)
2. (5x^{3}-125x)
3. ((a^{2}-25))
4. (x^{2}-12x + 27)
5. (10x^{2}+50x)
6. (16x^{2}-36)
7. the area of a rectangle can be represented by the expression (x^{2}+8x + 12). what do the factors of this expression represent in the context of this problem?
8. given the quadratic equation in factored form: (f(x)=(x - 2)(x + 4)). convert (f(x)) to standard form.

Explanation:

1. Factor \(3p^{2}+9p - 210\)

Step1: Factor out the greatest common factor (GCF)

The GCF of \(3p^{2}\), \(9p\) and \(-210\) is \(3\).

$$3p^{2}+9p - 210=3(p^{2}+3p - 70)$$

Step2: Factor the quadratic trinomial

We need to find two numbers \(m\) and \(n\) such that \(m + n=3\) and \(m\times n=- 70\). The numbers are \(10\) and \(-7\).

$$p^{2}+3p - 70=(p + 10)(p-7)$$

So, \(3p^{2}+9p - 210 = 3(p + 10)(p - 7)\)

2. Factor \(5x^{3}-125x\)

Step1: Factor out the GCF

The GCF of \(5x^{3}\) and \(-125x\) is \(5x\).

$$5x^{3}-125x=5x(x^{2}-25)$$

Step2: Use the difference - of - squares formula \(a^{2}-b^{2}=(a + b)(a - b)\)

Here \(a = x\) and \(b = 5\), so \(x^{2}-25=(x + 5)(x - 5)\)
So, \(5x^{3}-125x=5x(x + 5)(x - 5)\)

3. Factor \(a^{2}-25\)

Step1: Use the difference - of - squares formula

Here \(a^{2}-25\) with \(a=a\) and \(b = 5\)

$$a^{2}-25=(a + 5)(a - 5)$$

4. Factor \(x^{2}-12x + 27\)

Step1: Find two numbers \(m\) and \(n\)

We need \(m + n=-12\) and \(m\times n = 27\). The numbers are \(-3\) and \(-9\)

$$x^{2}-12x + 27=(x-3)(x - 9)$$

5. Factor \(10x^{2}+50x\)

Step1: Factor out the GCF

The GCF of \(10x^{2}\) and \(50x\) is \(10x\)

$$10x^{2}+50x=10x(x + 5)$$

6. Factor \(16x^{2}-36\)

Step1: Factor out the GCF

The GCF of \(16x^{2}\) and \(-36\) is \(4\)

$$16x^{2}-36=4(4x^{2}-9)$$

Step2: Use the difference - of - squares formula

Here \(a = 2x\) and \(b = 3\), so \(4x^{2}-9=(2x + 3)(2x-3)\)
So, \(16x^{2}-36=4(2x + 3)(2x - 3)\)

7. Analyze the area of the rectangle

Step1: Recall the formula for the area of a rectangle

The area of a rectangle \(A=l\times w\), where \(A=x^{2}+8x + 12\)

Step2: Factor the quadratic trinomial

We need \(m + n=8\) and \(m\times n = 12\). The numbers are \(6\) and \(2\)

$$x^{2}+8x + 12=(x + 6)(x + 2)$$

The factors \((x + 6)\) and \((x + 2)\) represent the length and the width of the rectangle.

8. Convert \(f(x)=(x - 2)(x + 4)\) to standard form

Step1: Use the FOIL method

$$
LATEXBLOCK0
$$

Answer:

1. \(3(p + 10)(p - 7)\)
2. \(5x(x + 5)(x - 5)\)
3. \((a + 5)(a - 5)\)
4. \((x-3)(x - 9)\)
5. \(10x(x + 5)\)
6. \(4(2x + 3)(2x - 3)\)
7. The factors represent the length and width of the rectangle.
8. \(x^{2}+2x - 8\)