## Math 8 Chapter 1 Lesson 2: Multiply polynomials by polynomials

## 1. Theoretical Summary

**– Rule: **To multiply a polynomial by a polynomial, we multiply each term of one polynomial by each term of the other polynomial and add the products together.

– In general, with A+B and C+D being two polynomials, the product (A+B)(C+D) is calculated by the following formula:

(A+B)(C+D)=AC+AD+BC+BD

**– Comment:** The product of two polynomials is a polynomial.

## 2. Illustrated exercise

**Question 1. **Calculate

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

b.\((2{x^2} – 1)({x^3} + 2x)\)

**Solution guide**

**Question a:**

\(\begin{array}{l} ({x^2} + 2x)(x + 3)\\ = ({x^2})(x + 3) + (2x)(x + 3)\\ = ({x^2})x + ({x^2})(3) + (2x)(x) + (2x)(3)\\ = {x^3} + 3{x^2} + 2{x^2} + 6x\\ = {x^3} + 5{x^2} + 6x \end{array}\)

**Sentence b:**

\(\begin{array}{l} (2{x^2} – 1)({x^3} + 2x)\\ = (2{x^2})({x^3} + 2x) + ( – 1)({x^3} + 2x)\\ = (2{x^2})({x^3}) + (2{x^2})(2x) – {x^3} – 2x\\ = 2{x^5} + 4{x^3} – {x^3} – 2x\\ = 2{x^5} + 3{x^3} – 2x \end{array}\)

**Verse 2.** Calculate

a.\((x + y)({x^2} – 3{y^3})\)

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

**Solution guide**

**Question a:**

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

**Sentence b:**

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

**Verse 3. **Reduce the expression \((x + y)(x – y)({x^2} + {y^2})\)

**Solution guide**

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

## 3. Practice

### 3.1. Essay exercises

**Question 1.** Calculate

a. \(({x^3} + 3x)(2x + 5)\)

b. \((3{x^3} – 2)({x^4} + 3{x^2})\)

**Verse 2.** Calculate

a. \(({x^2} + {y^2})({x^3} – 4{y^4})\)

b. \(({x^3} + 3{x^2}{y^2})(2{y^3} + {x^2}{y^4})\)

**Verse 3. **Collapse expression \((x + y)({x^2} – xy + {y^2})({x^3} – {y^3})\)

### 3.2. Multiple choice exercises

**Question 1: **The expression \(\left( {\frac{1}{2}{x^4} + 3} \right)\left( { – 1 + {x^3}} \right)\) has degree of?

A. 5

B. 6

C. 7

D. 8

**Verse 2: **Find b knowing \(\left( {2b + 32} \right)\left( {2b – 8} \right) = 0\). The search value of b is:

A. -16, 4

B. 16, -4

C. 4; -18

D. 18; 4

**Question 3: **The value of the expression \(- \left( {\frac{1}{2}b – 2} \right)\left( {b + 1} \right)\) at b=2 is

A. 1

B. 2

C. 3

D. 4

**Question 4: **Calculate

\(\left( {4{x^2} – \frac{1}{2}} \right)\left( {16{x^4} + 2{x^2} + \frac{1}{4 }} \right)\)

A. \(64{x^6} – \frac{1}{8}\)

B. \(64x^2-12\)

C. \(24x^2+1\)

D. \(5x^3+12\)

**Question 5: **Find x: \(x\left( {x + 1} \right) – {x^2} + 8 = 0\)

A. x = 2

B. x = 4

C. x = 6

D. x = 8

## 4. Conclusion

Through this lesson, you should achieve the following goals:

- Know the rules for multiplying polynomials by polynomials.
- Multiply polynomials by polynomials.

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