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**Solving Exercises Lesson 25 One-variable Polynomials (Chapter 7 Math 7 Connection)**

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### Solve problem 7.5 page 30 Math textbook 7 Connecting knowledge volume 2 – KNTT

a) Calculate \(\left( {\dfrac{1}{2}{x^3}} \right).\left( {4{x^2}} \right)\). Find the coefficient and degree of the obtained monomial.

b) Calculate \(\dfrac{1}{2}{x^3} – \dfrac{5}{2}{x^3}\). Find the coefficient and degree of the obtained monomial.

## Detailed instructions for solving Problem 7.5

**Solution method**

Step 1: Collapse

a) To multiply two mononomials, we multiply the two coefficients together and multiply the two powers of the variable together

b) To subtract two mononomials of the same degree, we subtract the coefficients together, keeping the power of the variable.

Step 2:

The monomial takes the form of the product of a real number to a power of the variable.

The real number is called the coefficient

The exponent of the power of the variable is called the degree of the monomial

**Detailed explanation**

a) \(\left( {\dfrac{1}{2}{x^3}} \right).\left( {4{x^2}} \right) = \left( {\dfrac{1} {2}.4} \right).\left( {{x^3}. {x^2}} \right) = 2. {x^5}\).

Coefficient: 2

Grade: 5

b) \(\dfrac{1}{2}{x^3} – \dfrac{5}{2}{x^3} = \left( {\dfrac{1}{2} – \dfrac{5} {2}} \right){x^3} = \dfrac{{ – 4}}{2}. {x^3} = – 2{x^3}\)

Coefficient: -2

Grade: 3

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### Solve lesson 7.6, page 30, Math 7 textbook Connecting knowledge volume 2 – KNTT

Given two polynomials:

\(\begin{array}{l}A = {x^3} + \dfrac{3}{2}x – 7{x^4} + \dfrac{1}{2}x – 4{x^2 } + 9\\B = {x^5} – 3{x^2} + 8{x^4} – 5{x^2} – {x^5} + x – 7\end{array}\)

a) Collapse and sort the above two polynomials according to the decreasing power of the variable.

b) Find the degree, highest coefficient and coefficient of freedom for each given polynomial.

## Detailed instructions for solving problems 7.6

**Solution method**

a) Step 1: Add and subtract mononomials of the same degree to get a reduced polynomial that contains no two mononomials of the same degree

Step 2: Sort the above polynomial according to the decreasing power of the variable.

b) + Degree of the polynomial is the degree of the term with the highest degree

+ The highest coefficient is the coefficient of the term with the highest degree

+ The coefficient of freedom is the coefficient of the term 0.

**Detailed explanation**

a)

\(\begin{array}{l}A(x) = {x^3} + \dfrac{3}{2}x – 7{x^4} + \dfrac{1}{2}x – 4{ x^2} + 9\\ = – 7{x^4} + {x^3} – 4{x^2} + \left( {\dfrac{3}{2}x + \dfrac{1}{ 2}x} \right) + 9\\ = – 7{x^4} + {x^3} – 4{x^2} + 2x + 9\\B(x) = {x^5} – 3 {x^2} + 8{x^4} – 5{x^2} – {x^5} + x – 7\\ = \left( {{x^5} – {x^5}} \right ) + 8{x^4} + \left( { – 3{x^2} – 5{x^2}} \right) + x – 7\\ = 0 + 8{x^4} + ( – 8 {x^2}) + x – 7\\ = 8{x^4} – 8{x^2} + x – 7\end{array}\)

b) * Polynomial A(x):

+ Degree of the polynomial is: 4

+ The highest coefficient is: -7

+ The coefficient of freedom is: 9

* Polynomial B(x):

+ Degree of the polynomial is: 4

+ The highest coefficient is: 8

+ The coefficient of freedom is: -7

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### Solving problem 7.7, page 30, Math 7 Textbook Connecting knowledge volume 2 – KNTT

Given two polynomials:

\(\begin{array}{l}P(x) = 5{x^3} + 2{x^4} – {x^2} + 3{x^2} – {x^3} – 2{ x^4} – 4{x^3}\\Q(x) = 3x – 4{x^3} + 8{x^2} – 5x + 4{x^3} + 5\end{array}\ )

a) Collapse and sort the above two polynomials according to the decreasing power of the variable.

b) Find the degree, highest coefficient and coefficient of freedom for each given polynomial.

## Detailed instructions for solving Problem 7.7

**Solution method**

a) Step 1: Add and subtract mononomials of the same degree to get a reduced polynomial that contains no two mononomials of the same degree

Step 2: Sort the above polynomial according to the decreasing power of the variable.

b) Replace each x value into P(x), Q(x) has been reduced and calculated.

**Detailed explanation**

a)

\(\begin{array}{l}P(x) = 5{x^3} + 2{x^4} – {x^2} + 3{x^2} – {x^3} – 2{ x^4} – 4{x^3}\\ = \left( {2{x^4} – 2{x^4}} \right) + \left( {5{x^3} – {x^ 3} – 4{x^3}} \right) + \left( { – {x^2} + 3{x^2}} \right)\\ = 0 + 0 + 2{x^2}\\ = 2{x^2}\\Q(x) = 3x – 4{x^3} + 8{x^2} – 5x + 4{x^3} + 5\\ = \left( { – 4{ x^3} + 4{x^3}} \right) + 8{x^2} + \left( {3x – 5x} \right) + 5\\ = 0 + 8{x^2} + ( – 2x) + 5\\ = 8{x^2} – 2x + 5\end{array}\)

b) P(1) = 2.1^{2} = 2

P(0) = 2.0^{2} = 0

Q(-1) = 8.(-1)^{2} – 2.(-1) +5 = 8 +2 +5 =15

Q(0) = 8.0^{2} – 2.0 + 5 = 5

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### Solve problem 7.8, page 30 Math 7 Textbook Connecting knowledge volume 2 – KNTT

Two pumps are used to pump water into a water tank. The first machine pumps 22 m . per hour^{3} water. The second pump pumps 16 m . per hour^{3} water. After both machines run for x hours, the first machine is turned off and the second machine runs for another 0.5 hours, the water tank is full.

Write the polynomial (x variable) representing the tank capacity (m .)^{3}). Know that before pumping, in the tank there are 1.5 m^{3} water. Find the highest coefficient and the coefficient of freedom of that polynomial.

## Detailed instructions for solving Problem 7.8

**Solution method**

Step 1: Write a polynomial expressing tank capacity = Amount of water 2 pumps in x hours + amount of tap water 2 pumps in 0.5 hours + Amount of water available in the tank

Step 2: Collapse the polynomial

+ The highest coefficient is the coefficient of the term with the highest degree

+ The coefficient of freedom is the coefficient of the term 0.

**Detailed explanation**

Polynomial V(x) = 22.x + 16.x + 0.5.16 + 1.5 = (22+16).x + 8 + 1.5 = 38.x + 9.5

Highest coefficient: 38

Factor of freedom: 9.5

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### Solve problem 7.9 page 30 Math textbook 7 Connecting knowledge volume 2 – KNTT

Write a polynomial F(x) that satisfies the following conditions at the same time:

The degree of F(x) is 3

The coefficient of x^{2} is equal to the factor of x and is equal to 2

The highest coefficient of F(x) is -6 and the free factor is 3.

## Detailed instructions for solving problem 7.9

**Solution method**

Write a polynomial that satisfies the following requirements:

The degree of a polynomial is the degree of the term with the highest degree

+ The highest coefficient is the coefficient of the term with the highest degree

+ The coefficient of freedom is the coefficient of the term 0.

**Detailed explanation**

F(x) = -6x^{3} + 2x^{2} + 2x + 3

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### Solution 7.10, page 30 Math 7 Textbook Connecting knowledge volume 2 – KNTT

Give it a check:

a) \(x = – \dfrac{1}{8}\) is the solution of the polynomial P(x) = 4x + \(\dfrac{1}{2}\)?

b) Of the three numbers 1; -1 and 2, which number is the solution of the polynomial Q(x) = x^{2} + x – 2 ?

## Detailed instructions for solving problems 7.10

**Solution method**

a) Replace the value \(x = – \dfrac{1}{8}\) into the polynomial P(x) = 4x + \(\dfrac{1}{2}\) to calculate the value P(\( – \dfrac{1}{8}\)). If P(\( – \dfrac{1}{8}\)) = 0, then \(x = – \dfrac{1}{8}\) is a solution of P(x)

b) Find Q(1); Q(-1); Q(2). At what value of x does Q(x) = 0 then that number is the solution of Q(x)

**Detailed explanation**

a) We have: P(\( – \dfrac{1}{8}\)) = 4.(\( – \dfrac{1}{8}\))+ \(\dfrac{1}{2} \)= (-\(\dfrac{1}{2}\)) + \(\dfrac{1}{2}\) = 0

So \(x = – \dfrac{1}{8}\) is the solution of the polynomial P(x) = 4x + \(\dfrac{1}{2}\)

b) Q(1) = 1^{2} +1 – 2 = 0

Q(-1) = (-1)^{2} + (-1) – 2 = -2

Q(2) = 2^{2} + 2 – 2 = 4

Since Q(1) = 0, x = 1 is the solution of Q(x)

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### Solve problem 7.11 page 30 Math textbook 7 Connecting knowledge volume 2 – KNTT

Mother gave Quynh 100 thousand dong. Quynh bought a set of learning tools for 37,000 VND and a Math reference book for x (thousand VND).

a) Find the polynomial (variable x) representing the amount of Quynh remaining (unit: thousand VND). Find the degree of that polynomial.

b) After buying the book, Quynh has just spent all the money her mother gave her. What is the price of the book?

## Detailed instructions for solving problem 7.11

**Solution method**

Write a polynomial that represents the remaining amount = the amount given by the mother – the amount purchased

Degree of a polynomial is the degree of the term with the highest degree

When all the money is spent, the remaining amount is equal to 0

**Detailed explanation**

a) Polynomial C(x) = 100 – 37 – x = – x + 63

Degree of polynomial is 1

b) After buying the book, we have 0 money left or – x + 63 = 0

\( \Rightarrow 63 = x\) or x = 63

So the price of the book is 63,000 dong

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