Question paper, Set 5

More than means add.

Symbolic form: $m + 4$

Answer: $m + 4$

$-10abm^2 = -10 \times a \times b \times m \times m$

Each number and each letter in the product is a factor.

Factors: $-10$, $a$, $b$, $m$, $m$

Answer: $-10$, $a$, $b$, $m$, $m$

Like terms have the same literal factors.

$\displaystyle -\frac{3mn^2}{4}$, $2n^2m$, $-5mn^2$ all have the literal factors $mn^2$ (note $n^2m = mn^2$).

$7m^2n$ and $2mn$ have different literal factors, so they are unlike.

Like terms: $\displaystyle -\frac{3mn^2}{4}$, $2n^2m$, $-5mn^2$

Answer: $\displaystyle -\frac{3mn^2}{4}$, $2n^2m$, $-5mn^2$

$3m + m - 6m - 5$

$= (3 + 1 - 6)m - 5$

$= -2m - 5$

Answer: $-2m - 5$

$\displaystyle \frac{4}{3}yz$ is one term, and the powers of its variables are whole numbers.

So it is a monomial.

Answer: Monomial

Degree of a monomial = sum of the powers of its variables.

$4 + 4 + 2 = 10$

So the degree of $\displaystyle \frac{2}{3}a^4b^4c^2$ is $10$.

Answer: $10$

Degrees of the terms: $9z^3$: $3$, $-5y^5z^2$: $7$, $-7yz$: $2$, $5y$: $1$.

Greatest among $\{3, 7, 2, 1\}$ is $7$.

So the degree of the polynomial is $7$.

Answer: $7$

Put $x = -3$:

$(-3)^3 + 2(-3)$

$= -27 - 6$

$= -33$

Answer: $-33$

$\displaystyle -\frac{5}{2}q^2 + qr + \frac{5}{2}r^2 + 4q^2 + \frac{5}{4}qr - \frac{3}{2}r^2$

$\displaystyle = (-\frac{5}{2} + 4)q^2 + (1 + \frac{5}{4})qr + (\frac{5}{2} - \frac{3}{2})r^2$

For $q^2$: $\displaystyle -\frac{5}{2} + 4 = \frac{-5 + 8}{2} = \frac{3}{2}$

For $qr$: $\displaystyle 1 + \frac{5}{4} = \frac{4 + 5}{4} = \frac{9}{4}$

For $r^2$: $\displaystyle \frac{5}{2} - \frac{3}{2} = \frac{5 - 3}{2} = \frac{2}{2} = 1$

$\displaystyle = \frac{3}{2}q^2 + \frac{9}{4}qr + r^2$

Answer: $\displaystyle \frac{3}{2}q^2 + \frac{9}{4}qr + r^2$

Write like terms one below the other and add each column.

$$\begin{array}{rrr} 9x & +5t & \\ 6x & & +2y \\ & 7t & +3y \\ \hline 15x & +12t & +5y \end{array}$$

Sum $= 15x + 12t + 5y$

Answer: $15x + 12t + 5y$

$(5p - 9r) - (6p - 8r)$

$= 5p - 9r - 6p + 8r$

$= (5 - 6)p + (-9 + 8)r$

$= -p - r$

Answer: $-p - r$

(i) $\displaystyle 4(7x^2 - \frac{1}{4}xy - 6y^2) - (2x^2 + \frac{2}{3}xy + \frac{1}{3}y^2)$

$\displaystyle = 28x^2 - xy - 24y^2 - 2x^2 - \frac{2}{3}xy - \frac{1}{3}y^2$

$\displaystyle = (28 - 2)x^2 + (-1 - \frac{2}{3})xy + (-24 - \frac{1}{3})y^2$

For $xy$: $\displaystyle -1 - \frac{2}{3} = \frac{-3 - 2}{3} = -\frac{5}{3}$

For $y^2$: $\displaystyle -24 - \frac{1}{3} = \frac{-72 - 1}{3} = -\frac{73}{3}$

$\displaystyle = 26x^2 - \frac{5}{3}xy - \frac{73}{3}y^2$

(ii) $\displaystyle (7x^2 - \frac{1}{4}xy - 6y^2) - 3(2x^2 + \frac{2}{3}xy + \frac{1}{3}y^2)$

$\displaystyle = 7x^2 - \frac{1}{4}xy - 6y^2 - 6x^2 - 2xy - y^2$

$\displaystyle = (7 - 6)x^2 + (-\frac{1}{4} - 2)xy + (-6 - 1)y^2$

For $xy$: $\displaystyle -\frac{1}{4} - 2 = \frac{-1 - 8}{4} = -\frac{9}{4}$

$\displaystyle = x^2 - \frac{9}{4}xy - 7y^2$

Answer: (i) $\displaystyle 26x^2 - \frac{5}{3}xy - \frac{73}{3}y^2$ (ii) $\displaystyle x^2 - \frac{9}{4}xy - 7y^2$

Write $3m^3 - 3n^3 + 5$ on top and the expression to be subtracted below it, like terms in the same column.

$$\begin{array}{rrrr} 3m^3 & -3n^3 & +5 & \\ 3m^3 & +3n^3 & +8 & +6mn \\ (-) & (-) & (-) & (-) \\ \hline & -6n^3 & -3 & -6mn \end{array}$$

Change the sign of every term in the lower row, then add each column.

The $m^3$ terms cancel.

$= -6n^3 - 3 - 6mn$

Answer: $-6n^3 - 3 - 6mn$

$= (8 \times (-4))(p^2 \times p)q^2$

$= -32p^3q^2$

Answer: $-32p^3q^2$

$= (2x^2)(-x^2y) + (2x^2)(x) + (2x^2)(-5xy)$

$= -2x^4y + 2x^3 - 10x^3y$

Answer: $-2x^4y + 2x^3 - 10x^3y$

$= 6a^2 + 6a - 10a - 10$

$= 6a^2 - 4a - 10$

Answer: $6a^2 - 4a - 10$

$= (9p^2 - 12pq + 12pr + 12pq - 16q^2 + 16qr) - (12pr - 16qr)$

$= 9p^2 - 12pq + 12pr + 12pq - 16q^2 + 16qr - 12pr + 16qr$

$= 9p^2 + (-12 + 12)pq + (12 - 12)pr - 16q^2 + (16 + 16)qr$

$= 9p^2 - 16q^2 + 32qr$

Answer: $9p^2 - 16q^2 + 32qr$

$7xy(x - 2y) - 7xy(3x + 2y)$

$= 7x^2y - 14xy^2 - 21x^2y - 14xy^2$

$= (7 - 21)x^2y + (-14 - 14)xy^2$

$= -14x^2y - 28xy^2$

Answer: $-14x^2y - 28xy^2$

$= 3p^2r - 3pr^2 - 8p^2r - 4pr^2 - 6p^2r + 6p^2r^2 + 2p^2r - 2pr^2$

$= (3 - 8 - 6 + 2)p^2r + (-3 - 4 - 2)pr^2 + 6p^2r^2$

$= -9p^2r - 9pr^2 + 6p^2r^2$

Answer: $-9p^2r - 9pr^2 + 6p^2r^2$

Remove the innermost brackets first.

$= 5x + [y - \{3t - 2x - 2t\}]$

$= 5x + [y - \{t - 2x\}]$

$= 5x + [y - t + 2x]$

$= 5x + y - t + 2x$

$= 7x + y - t$

Answer: $7x + y - t$