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Solve the exercises Joint practice page 44 (Chapter 7 Math 7 Connect)
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Exercise 7.36 page 45 math 7 volume 2 KNTT
Simplify the following expression:
$(5x^3 – 4x^2) : 2x^2 + (3x^4 + 6x) : 3x – x(x^2 – 1)$
Solution guide:
$(5x^3 – 4x^2) : 2x^2 + (3x^4 + 6x) : 3x – x(x^2 – 1)$
$= (5x^3 : 2x^2) + (-4x^2 : 2x^2) + (3x^4 : 3x) + (6x : 3x) + (-x . x^2) + (-x . (-1))$
$= \frac{5}{2}x – 2 + x^3 + 2 – x^3 + x$
$= (x^3 – x^3) + (\frac{5}{2}x + x) + (-2 + 2)$
$= \frac{7}{2}x$
Exercise 7.37 page 45 math 7 volume 2 KNTT
Simplify the following expressions:
a) $2x(x + 3) – 3x^2(x + 2) + x(3x^2 + 4x – 6)$.
b) $3x(2x^2 – x) – 2x^2(3x + 1) + 5(x^2 – 1)$
Solution guide:
a) $2x(x + 3) – 3x^2(x + 2) + x(3x^2 + 4x – 6)$.
$= 2x.x + 2x.3 – 3x^2 . x – 3x^2 . 2 + x . 3x^2 + x . 4x + x . (-6)$
$= 2x^2 + 6x – 3x^3 – 6x^2 + 3x^3 + 4x^2 – 6x$
$=0$
b) $3x(2x^2 – x) – 2x^2(3x + 1) + 5(x^2 – 1)$
$= 3x. 2x^2 – 3x.x – 2x^2 . 3x – 2x^2 . 1 + 5.x^2 – 5.1$
$= 6x^3 – 3x^2 – 6x^3 – 2x^2 + 5x^2 – 5$
$= -5$
Exercise 7.38, page 45, math 7 episode 2 KNTT
Find the value of $x$, knowing that:
a) $3x^2 – 3x(x – 2) = $36.
b) $5x(4x^2 – 2x + 1) – 2x(10x^2 – 5x + 2) = -36$.
Solution guide:
a) $3x^2 – 3x(x – 2) = $36.
$3x^2 – (3x.x – 3x.2) = $36
$3x^2 – 3x^2 + 6x = $36
$6x = 36$
=> $x = 6$
b) $5x(4x^2 – 2x + 1) – 2x(10x^2 – 5x + 2) = -36$.
$5x.4x^2 + 5x.(-2x) + 5x.1 – (2x.10x^2 + 2x.(-5x) + 2x.2) = -36$
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$20x^3 – 10x^2 + 5x – (20x^3 – 10x^2 + 4x) = -36$
$20x^3 – 10x^2 + 5x – 20x^3 + 10x^2 – 4x = -36$
$x = -36$
Exercise 7.39 page 45 math 7 volume 2 KNTT
Do the following calculations:
a) $(x^3 – 8) : (x – 2)$
b) $(x – 1)(x + 1)(x^2 + 1)$
Solution guide:
a) $(x^3 – 8) : (x – 2)$
b) $(x – 1)(x + 1)(x^2 + 1)$
$= (xx + x.1 – 1.x – 1.1)(x^2 + 1)$
$= (x^2 + x – x – 1)(x^2 + 1)$
$=(x^2 – 1)(x^2 + 1)$
$= x^4 – 1$
Exercise 7.40 page 45 math 7 episode 2 KNTT
In a game at the Math club, the game owner wrote on the board the expression:
$P(x) = x^2(7x – 5) – (28x^5 – 20x^4 – 12x^3) : 4x^2$.
The rule of the game is that after the master reads a certain number a, the teams must calculate the value of $P(x)$ at $x = a$. The team that calculates correctly and calculates the fastest wins.
When the game owner just read $a = 5$, Square immediately calculated $P(a) = 15$ and won. Do you know how Vuong does it?
Solution guide:
Square shortens the previous problem so that the polynomial $P(x)$ is neater and easier to calculate mentally.
$P(x) = x^2(7x – 5) – (28x^5 – 20x^4 – 12x^3) : 4x^2$.
$P(x) = x^2 . 7x – x^2 . 5 – (28x^5 : 4x^2 – 20x^4 : 4x^2 – 12x^3 : 4x^2)$
$P(x) = 7x^3 – 5x^2 – 7x^3 + 5x^2 + 3x$
$P(x) = 3x$
* So: when the game owner reads $a = 5$, Square just needs to replace $a = 5$ into the expression $P(x) = 3x$ to easily calculate: $P(3) = 3 . 5 = 15$
Exercise 7.41 page 45 math 7 volume 2 KNTT
Find the number $b$ such that the polynomial $x^3 – 3x^2 + 2x – b$ is divisible by the polynomial $x – 3$.
Solution guide:
* Set calculation:
=> Thus: The remainder in the above polynomial is $– b + 6$.
For polynomial $x^3 – 3x^2 + 2x – b$ to be divisible by polynomial $x – 3$, the remainder is 0.
$– b + 6 = 0$
=> $b = 6$
