## Math 8 Chapter 1 Lesson 8: Factoring polynomials using the term grouping method

## 1. Theoretical Summary

– In a problem sometimes the common factor will not appear, but is “hidden” in the problem, so we need to perform some transformations so that the common factor appears.

**Eg:**

\(\begin{array}{l} {x^2} – xy + 9x – 9y\\ = ({x^2} – xy) + (9x – 9y)\\ = x(x – y) + 9 (x – y)\\ = (x + 9)(x – y) \end{array}\)

– During the test, in some problems, students have to change the sign of the polynomial to appear the common factor

**– Note: **A =-(-A)

## 2. Illustrated exercise

**Question 1:** Factorize the polynomial: \({x^2} – 2xy – 5x + 10y\)

**Solution guide**

\(\begin{array}{l} {x^2} – 2xy – 5x + 10y\\ = \left( {{x^2} – 2xy} \right) – \left( {5x – 10y} \right )\\ = x(x – 2y) – 5(x – 2y)\\ = (x – 5)(x – 2y) \end{array}\)

**Verse 2: **Factor the following polynomial \({x^3} + y(1 – 3{x^2}) + x(3{y^2} – 1) – {y^3}\)

**Solution guide**

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

**Question 3:** Factorize the polynomial \({x^3}z + {x^2}yz – {x^2}{z^2} – xy{z^2}\)

**Solution guide**

\(\begin{array}{l} {x^3}z + {x^2}yz – {x^2}{z^2} – xy{z^2}\\ = ({x^3} z – {x^2}{z^2}) + ({x^2}yz – xy{z^2})\\ = {x^2}z(x – z) + xyz(x – z) \\ = ({x^2}z + xyz)(x – z) \end{array}\)

## 3. Practice

### 3.1. Essay exercises

**Question 1**: Factorize polynomials

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

b. \(2{{x}^{2}}-3xy-6x+9y\)

**Verse 2:** Factor the following polynomial \({{x}^{2}}+x(2{{y}^{2}}+y)+y(x-2{{x}^{2}} )-{{y}^{2}}\)

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

### 3.2. Multiple choice exercises

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

A. \((x+y)(x-5)\)

B. \((x+y)(x+5)\)

C. \((xy)(x-5)\)

D. \((x+5y)(xy)\)

**Verse 2: **Given the polynomial \(4x^2-y^2+10y-25\). Which of the following results can be obtained by factoring the polynomial?

A. \((2x+y-5)(2x-y+5)\)

B. \((2x-y-5)(2x-y+5)\)

C. \((x+y-5)(x-y+5)\)

D. \((x+2y-5)(x-2y+5)\)

**Question 3: **Quickly calculate the expression value \(P=11,9.0,6+11,9.0,4\) we get which of the following results?

A. P=110

B. P=11

C. P=1.1

D. P=0.11

**Question 4: **Calculate \(56^2+31^2-13^2+56.62\)

A. 470

B. 4700

C. 7400

D. 740

**Question 5: **Find x knowing \({x^3} – {x^2} – 25x + 25 = 0\)

A. x = 1; x = 5; x= -5

B. x = 1; x = 5

C. x = -5

D. x= -1; x= -5 ; x = 5

## 4. Conclusion

Through this lesson, you should know the following:

- Understand how to factor polynomials.
- Apply the term group method to solve some related problems.

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