## Detailed instructions for solving Problem 7.33

**Solution method**

To divide a polynomial by a monomial, we divide each term of the polynomial by the monomial and then sum the results.

**Detailed explanation**

a) (0.5x^{5} + 3.2x^{3} – 2x^{2} ) : 0.25x^{2}

= 0.5x^{5} : 0.25x^{2} + 3.2x^{3} : 0.25x^{2} + (2x^{2} : 0.25x^{2})

= (0.5:0.25).(x^{5} : x^{2}) + (3.2 : 0.25). (x^{3} : x^{2} ) + (2 : 0.25). (x^{2} : x^{2})

= 2x^{3} + 12.8x + 4

b) (0.5x^{5} + 3.2x^{3} – 2x^{2} ) : 0.25x^{3}

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### Solution 7.34 page 43 Math textbook 7 Connecting knowledge volume 2 – KNTT

In each of the following cases, find the quotient Q(x) and the remainder R(x) in dividing F(x) by G(x) and express F(x) as:

F(x) = G(x) . Q(x) + R(x)

a) F(x) = 6x^{4} – 3x^{3} + 15x^{2} + 2x – 1 ; G(x) = 3x^{2}

b) F(x) = 12x^{4} + 10x^{3} – x – 3 ; G(x) = 3x^{2} + x + 1

## Detailed instructions for solving problem 7.34

**Solution method**

+) To divide polynomial A by polynomial B, we do the following:

Step 1: Set the divisibility similar to dividing two natural numbers. Divide the highest-order term of A by the highest-order term of B.

Step 2: Take A minus the product of B with the new quotient obtained in step 1

Step 3: Divide the highest order term of the first remainder by the highest term of B

Step 4: Subtract the product B from the first remainder with the quotient obtained in step 3

Step 5: Do the same as above

When the last remainder is of degree less than the degree of B, the division ends.

+) Write A = B. Q + R

**Detailed explanation**

a)

quotient Q(x) = 2x^{2} – x + 5

Residual R(x) = 2x – 1

We have: F(x) = 3x^{2} . (2x .)^{2} – x + 5) + 2x – 1

b)

Quotient Q(x) = 4x^{2} + 2x – 2

Residual R(x) = -x – 1

We have: F(x) = (3x^{2} + x + 1) . (4x .)^{2} + 2x – 2) – x – 1

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### Solve problem 7.35 page 43 Math textbook 7 Connecting knowledge volume 2 – KNTT

Tam is confused when he wants to find the quotient and remainder in the division of the polynomial 21x – 4 by 3x^{2} . Can I help you Tam?

## Detailed instructions for solving problem 7.35

**Solution method**

When the division polynomial has degree less than the degree of the divisor, the quotient is 0, the remainder is the divisor

**Detailed explanation**

Divide polynomial 21x – 4 by 3x^{2} quotient is 0, remainder 21x – 4

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