## Math 8 Chapter 1 Lesson 9: Factorizing polynomials by combining many methods

## 1. Theoretical Summary

For some problems, we cannot immediately apply the methods we have learned, but we must use a combination of many learned methods such as:

- Set the common factor.
- Use equality constants.
- Elemental group.

**Eg:** Factoring polynomials:

\(\begin{array}{l} {x^3} – 4x + 4x\\ = x({x^2} – 4x + 4)\\ = x{(x – 2)^2} \end{ array}\)

## 2. Illustrated exercise

**Question 1:** Factorize the polynomial: \(2{x^4} + 3{x^3} + 2{x^2} + 3x\)

**Solution guide**

\(\begin{array}{l} 2{x^4} + 3{x^3} + 2{x^2} + 3\\ = x(2{x^3} + 3{x^2} + 2x + 3)\\ = x\left[ {(2{x^3} + 3{x^2}) + (2x + 3)} \right]\\ = x\left[ {{x^2}(2x + 3) + (2x + 3)} \right]\\ = x({x^2} + 1)(2x + 3) \end{array}\)

**Verse 2:** Factoring polynomials

a) \( – 3{x^2} + 12x – 12 + 3{y^2}\)

b) \(16 + 4xy – {x^2} – 4{y^2}\)

**Solution guide**

**Question a:**

\(\begin{array}{l} – 3{x^2} + 12x – 12 + 3{y^2}\\ = – 3({x^2} – 4x + 4 – {y^2}) \\ = – 3\left[ {({x^2} – 4x + 4) – {y^2}} \right]\\ = – 3\left[ {{{(x – 2)}^2} – {y^2}} \right]\\ = – 3(x – 2 – y)(x – 2 + y) \end{array}\)

**Sentence b:**

\(\begin{array}{l} 16 + 4xy – {x^2} – 4{y^2}\\ = 16 – ({x^2} – 4xy + 4{y^2})\\ = 16 – {(x – 2y)^2}\\ = (4 – x + 2y)(4 + x – 2y) \end{array}\)

**Question 3: **Factor the polynomial \({x^2} – 6x + 8\)

**Solution guide**

\(\begin{array}{l} {x^2} – 6x + 8\\ = {x^2} – 6x + 9 – 1\\ = ({x^2} – 6x + 9) – 1\ \ = {(x – 3)^2} – 1\\ = (x – 3 – 1)(x – 3 + 1)\\ = (x – 4)(x – 2) \end{array}\)

### 3.1. Essay exercises

**Question 1: **Factoring polynomials

a) \({{x}^{4}}-3{{x}^{3}}+3{{x}^{2}}-3x+x\)

b) \(5{{x}^{4}}-2{{x}^{3}}-5{{x}^{2}}+2x\)

**Verse 2:** Factoring polynomials

a) \(-3{{x}^{2}}+6x-3+3{{y}^{2}}\)

b) \(9+6xy-9{{x}^{2}}-{{y}^{2}}\)

**Question 3:** Factorize the polynomial \({{x}^{2}}-4x+3\)

### 3.2. Multiple choice exercises

**Question 1: **Given the polynomial \(x^5-x^4y+2x^4-2x^3y\). Which of the following results can be obtained by factoring the polynomial?

A. \((x+y)(x-2)\)

B. \((xy)(x+2)\)

C. \(x^2(xy)(x+2)\)

D. \(x^3(xy)(x+2)\)

**Verse 2: **Given the polynomial \(a^2(bc)+b^2(ca)+c^2(ab)\). Which of the following results can be obtained by factoring the polynomial?

A. \((a-1)(b-1)(c-1)\)

B. \((1-a)(1-b)(1-c)\)

C. \((ab)(bc)(ca)\)

D. \((a+b)(b+c)(c+a)\)

**Question 3: **Given the expression \(P=(xy)(x^2-y^2-8x-8y)\). Calculate the value of the expression P at x=4; y=3

A. P=15

B. P=8

C. P=18

D. P=6

**Question 4: **Find x knowing \((x-3)(x^2-2x)+(2-x)(x-3)=0\)

A. x=1

B. x=2

C. x=3

D. x=1;2;3

**Question 5: **Find x knowing \(4{x^3} – 8{x^2} – 9x + 18 = 0\)

A. \(x=2\)

B. \(x = \pm 2\)

C. \(x = 2;x = \pm \frac{3}{2}\)

D. \(x = \frac{3}{2}\)

## 4. Conclusion

Through this lesson, you should know the following:

- Know how to combine the methods of polynomial analysis into the learned factors.
- Solve polynomial problems.

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